Pulse sequence design protocol

ABSTRACT

Systems and methods are disclosed for a pulse sequence that reduces disorder and/or interaction effects in spin systems. A protocol can be used to design a pulse sequence that includes altering the frame orientation of the spin system with each electromagnetic pulse in the pulse sequence. The frame orientations during the sequence can conform to certain conditions. The number positive rotations along each axis can be the same as the number negative rotations along the respective axis. The number of rotations along one axis should be the same as the number of rotations along the other axes.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority to U.S. ProvisionalApplication No. 62/814,775, entitled “Surpassing the Interaction Limitto Quantum Sensing with Fault-Tolerant Control,” filed on Mar. 6, 2019,the disclosure of which is hereby incorporated by reference in itsentirety.

STATEMENT OF GOVERNMENTAL INTEREST

This invention was made with Government support under Grant #N00014-15-1-2846 awarded by the United States Department of Defense andthe Office of Naval Research and Grant # W911NF-15-1-0548 awarded by theUnited States Army and the Army Research Office. The government hascertain rights in the invention.

TECHNICAL FIELD

The invention relates to spin systems, and more particularly to designprotocols to produce pulse sequences to control interacting spinensembles.

COPYRIGHT NOTICE

This patent disclosure may contain material that is subject to copyrightprotection. The copyright owner has no objection to the facsimilereproduction by anyone of the patent document or the patent disclosureas it appears in the U.S. Patent and Trademark Office patent file orrecords, but otherwise reserves any and all copyright rights.

BACKGROUND

The ability to robustly control and manipulate the dynamics of a quantumsystem, such as a single quantum spin, is an important aspect of devicesor methods that take advantage of quantum dynamics, such as ranging fromquantum metrology and quantum simulation. Increasing the accuracy androbustness of control of quantum dynamics can increase the sensitivityand number of applications of such quantum dynamics. However, robustcontrol and manipulation of quantum dynamics are limited by noise.

Among efforts to improve control of quantum dynamics, the application ofperiodic control pulses, sometimes referred to as Floquet driving, hasemerged as a tool for the control and engineering of quantum dynamics.One example of such approaches is dynamical decoupling of a target spinfrom its environment, in which a series of pulses manipulates a quantumspin and cancels the effects of interactions with environmental noise.The periodic structure of these pulse sequences can give rise tofrequency-selective resonances, enabling high sensitivity to externalsignals measured by the quantum system (e.g. magnetic fields).

In addition to manipulating the dynamics of a single spin and applyingsuch techniques to a plurality of spins, periodic control pulses alsoallow for control of interactions between spins, even when only globalcontrol (i.e., control over the plurality of spins without specificaddressing of individual spins) is available, as is typically the casefor large-scale quantum systems with many degrees of freedom. Oneexample is the WAHUHA pulse sequence and its extension MREV-8, each ofwhich can cancel the dipolar interaction between spin-½ particles toleading order.

SUMMARY

Systems and methods are disclosed for a pulse sequence that reducesdisorder and/or interaction effects in spin systems. A protocol can beused to design a pulse sequence that includes altering the frameorientation of the spin system with each electromagnetic pulse in thepulse sequence. The frame orientations during the sequence can conformto certain conditions. The number positive rotations along each axis canbe the same as the number negative rotations along the respective axis.The number of rotations along one axis should be the same as the numberof rotations along the other axes.

According to some embodiments, a method of reducing disorder andinteraction effects in a spin system includes: applying a sequence of nelectromagnetic pulses to the spin system, the spin system having aframe orientation in an evolution period τ₀ before a first pulse k=1 ofthe sequence of electromagnetic pulses; and altering the frameorientation of the spin system with each electromagnetic pulse in thesequence of pulses, each electromagnetic pulse being one or more of aπ/2 rotation or a π rotation, the frame orientations during the sequenceconforming to the following relations:

${{\sum\limits_{k}{F_{\mu\; k}\tau_{k}}} = 0},{and}$${{\sum\limits_{k}{{F_{xk}}\tau_{k}}} = {{\sum\limits_{k}{{F_{yk}}\tau_{k}}} = {\sum\limits_{k}{{F_{zk}}\tau_{k}}}}},$where F_(μk) represents the frame orientation of the spin system in arespective evolution period of duration τ_(k) after pulse k for eachspin direction μ=x, y, z for 0≤k≤n, and where k=0 corresponds to theframe orientation F_(μ0) in the evolution period before the first pulsek=1.

In some embodiments, the sequence of electromagnetic pulses is periodic,and the pulses are equally spaced.

In some embodiments, at least one electromagnetic pulse of the sequenceof electromagnetic pulses includes two or more π/2 rotations, and thespin system further comprises intermediary frame orientationsrepresenting the frame orientation of the spin system after each but afinal π/2 rotation, the intermediary frame orientations conforming tothe following relations:

${{\sum\limits_{k}F_{\mu\; v}} = 0},{{\sum\limits_{k}{F_{xv}}} = {{\sum\limits_{k}{F_{yv}}} = {\sum\limits_{k}{F_{zv}}}}}$where F_(μv) represents the frame orientation of the spin system foreach intermediary frame v for each spin direction μ=x, y, z.

In some embodiments, for each π rotation, the frame orientation of thespin system further comprises an intermediary frame orientationrepresenting the frame orientation of the spin system after the firstπ/2 rotation of the π rotation, intermediary frame orientations togetherconforming to the following relations:

${{\sum\limits_{k}F_{\mu\; v}} = 0},{{\sum\limits_{k}{F_{xv}}} = {{\sum\limits_{k}{F_{yv}}} = {\sum\limits_{k}{F_{zv}}}}}$where F_(μv) represents the frame orientation of the spin system foreach intermediary frame v for each spin direction μ=x, y, z.

In some embodiments, for each pair of axes μ,μ=x, y, z, the parity offrame changes sums to zero such that

${{{\sum\limits_{k}{F_{\mu,k}F_{v,{k + 1}}}} + {F_{\mu,{k + 1}}F_{v,k}}} = 0},$for (μ, v)=(x, y), (x, z), (y, z).

In some embodiments, the chirality of frame changes sums to zero suchthat the cyclic sum

${{\sum\limits_{k}{{\overset{\rightarrow}{F}}_{k} \times {\overset{\rightarrow}{F}}_{k + 1}}} = \overset{\rightarrow}{0}},$where {right arrow over (F)}_(k)=Σ_(μ)F_(μ,k){right arrow over (e)}_(μ)and {right arrow over (e)}_(μ) are the unit vectors along axisdirections.

In some embodiments, the method further includes generating an effectivemagnetic field {right arrow over (B)}_(eff) as seen by the driven spins;and initializing the frame orientation of the spin system to beperpendicular to the effective magnetic field.

In some embodiments, the sequence of electromagnetic pulses is used toincrease the coherence time of an ensemble of nitrogen-vacancy (NV)centers in diamond beyond a spin-spin interaction sensitivity limit.

In some embodiments, the sequence of electromagnetic pulses is used toincrease the coherence time of a magnetic field sensing ensemble ofnitrogen-vacancy (NV) centers in diamond such that a sensitivity of themagnetic field sensing ensemble of NV centers overcomes a spin-spininteraction sensitivity limit.

According to some embodiments, a system can include: a quantum sensorcomprising an ensemble of spins in solid state, the ensemble of spinshaving a density in which the interactions between the spins limit acoherence time of the ensemble of spins in solid state; and a pulsegenerator configured to apply electromagnetic radiation to the quantumsensor, the electromagnetic radiation decoupling the interactionsbetween the spins to increase the coherence time beyond a spin-spininteraction sensitivity limit of the ensemble of spins when measuring atarget signal.

In some embodiments, the quantum sensor comprising an ensemble of NVcenters in diamond of density r ppm, and the coherence time is increasedto be longer than a value of 72/r us (as determined from the scaling ofthe interaction limit), up to 1 ms.

In some embodiments, the pulse generator applies electromagneticradiation to the quantum sensor according to the method of claim 1.

According to some embodiments, a spin system can include: a pulsegenerator configured to a sequence of n electromagnetic pulses to thespin system, the spin system having a frame orientation in an evolutionperiod τ₀ before a first pulse k=1 of the sequence of electromagneticpulses, each electromagnetic pulse corresponding to a frame of thesequence of pulses, and each electromagnetic pulse being one or more ofa π/2 rotation or a π rotation, the frame orientations during thesequence conforming to the following relations:

${{\sum\limits_{k}{F_{\mu\; k}\tau_{k}}} = 0},{and}$${{\sum\limits_{k}{{F_{xk}}\tau_{k}}} = {{\sum\limits_{k}{{F_{yk}}\tau_{k}}} = {\sum\limits_{k}{{F_{zk}}\tau_{k}}}}},$where F_(μk) represents the frame orientation of the spin system in arespective evolution period of duration τ_(k) after pulse k for eachspin direction μ=x, y, z for 0≤k≤n, and where k=0 corresponds to theframe orientation F_(μ0) in the evolution period before the first pulsek=1.

In some embodiments, the sequence of electromagnetic pulses is periodic,and the pulses are equally spaced.

In some embodiments, at least one electromagnetic pulse of the sequenceof electromagnetic pulses includes two or more π/2 rotations, and thespin system further comprises intermediary frame orientationsrepresenting the frame orientation of the spin system after each but afinal π/2 rotation, the intermediary frame orientations conforming tothe following relations:

${{\sum\limits_{k}F_{\mu\; v}} = 0},{{\sum\limits_{k}{F_{xv}}} = {{\sum\limits_{k}{F_{yv}}} = {\sum\limits_{k}{F_{zv}}}}}$where F_(μv) represents the frame orientation of the spin system foreach intermediary frame v for each spin direction μ=x, y, z.

In some embodiments, for each π rotation, the frame orientation of thespin system further comprises an intermediary frame orientationrepresenting the frame orientation of the spin system after the firstπ/2 rotation of the π rotation, intermediary frame orientations togetherconforming to the following relations:

${{\sum\limits_{k}F_{\mu\; v}} = 0},{{\sum\limits_{k}{F_{xv}}} = {{\sum\limits_{k}{F_{yv}}} = {\sum\limits_{k}{F_{zv}}}}}$where F_(μv) represents the frame orientation of the spin system foreach intermediary frame v for each spin direction μ=x, y, z.

In some embodiments, for each pair of axes μ,μ=x, y, z, parity of framechanges experienced by the spin system sums to zero such that

${{{\sum\limits_{k}{F_{\mu,k}F_{v,{k + 1}}}} + {F_{\mu,{k + 1}}F_{v,k}}} = 0},$for (μ, v)=(x, y), (x, z), (y, z).

In some embodiments, the chirality of frame changes experienced by thespin system sums to zero such that the cyclic sum

${{\sum\limits_{k}{{\overset{\rightarrow}{F}}_{k} \times {\overset{\rightarrow}{F}}_{k + 1}}} = \overset{\rightarrow}{0}},$where {right arrow over (F)}_(k)=Σ_(μ)F_(μ,k){right arrow over (e)}_(μ)and {right arrow over (e)}_(μ) are the unit vectors along axisdirections.

These and other capabilities of the disclosed subject matter will bemore fully understood after a review of the following figures, detaileddescription, and claims. It is to be understood that the phraseology andterminology employed herein are for the purpose of description andshould not be regarded as limiting.

BRIEF DESCRIPTION OF THE FIGURES

For a more complete understanding of various embodiments of thedisclosed subject matter, reference is now made to the followingdescriptions taken in connection with the accompanying drawings, inwhich:

FIG. 1A is an illustrating of a spin ensemble quantum sensor, accordingto some embodiments;

FIG. 1B is an illustration of magnetic sublevels, according to someembodiments;

FIG. 2 is a graphical illustration of volume-normalized magnetic fieldsensitivity (η_v) as a function of spin density, ρ, according to someembodiments;

FIG. 3 views (a) through (c) illustrate example aspects of afault-tolerant sequence design, according to some embodiments;

FIG. 4 views (a) through (f) illustrate example pulse sequences,according to some embodiments;

FIG. 5A illustrates intermediate frame orientations, according to someembodiments;

FIG. 5B illustrates interacting picture Hamiltonian evolution during apulse, according to some embodiments;

FIG. 5C illustrates an example portion of a pulse sequence, according tosome embodiments;

FIG. 5D illustrates an example portion of a pulse sequence, according tosome embodiments;

FIG. 5E illustrates an example portion of a pulse sequence, according tosome embodiments;

FIG. 5F illustrates an example portion of a pulse sequence, according tosome embodiments;

FIG. 6, views (a) through (c) show example aspects of Seq. A and Seq. Bpulse sequences, according to some embodiments;

FIGS. 7A-7D show example dynamics and features of various pulsesequences, according to some embodiments;

FIGS. 8A-8D show example aspects of an example pulse sequence developedusing aspects of the disclosed protocol, according to some embodiments;

FIGS. 9A-9B show example aspects of XY-8 and Seq. B pulse sequences,according to some embodiments;

FIGS. 10A-10E show example aspects of implementations of fault-tolerantsequences using dense NV centers in diamond, according to someembodiments; and

FIGS. 11A-11E show example aspects of XY-8 and Seq. C pulse sequences,according to some embodiments.

DESCRIPTION

Quantum sensors, such as nitrogen vacancies in diamond, can be used todetect small fields, such as weak magnetic fields with, high sensitivityand/or precision. Such high sensitivity and/or precision sensing canenable myriad applications ranging from nanoscale nuclear magneticresonance spectroscopy of biomolecules to local probing of exoticcondensed matter phenomena. In some applications, the magnetic fieldsensitivity can be improved by increasing the density of sensors in agiven volume. However, increasing the density of sensors may not alwaysincrease sensitivity to the same degree beyond a critical density. Forexample, in some applications, beyond a critical density (or, moregenerally, as density increases) this improvement in sensitivity ishindered by interactions between the sensors themselves (e.g., between asensor and its nearest neighbors), which can be referred to as“interaction effects” or simply “interactions.” In some applications,sensitivity improvement can also or alternatively be hindered byincreasing inhomogeneity that can result from a higher sensor density,which can be referred to as “disorder” or “inhomogeneity.” Interactionsand disorder are described in more detail throughout the presentdisclosure, for example as described below in the Disorder andInteractions section. In some embodiments, control errors can also giverise to inhomogeneities in the response of individual sensors in theensemble and result in a further decrease in sensitivity.

Existing techniques for designing pulse sequences to be applied toquantum sensors do not address all of the above limitations onsensitivity and other important performance measures. Accordingly, thereis a need for a systematic design protocol for designing pulse sequencesthat can simultaneously addresses sensor-sensor interactions, on-sitedisorder, and control imperfections, in order to surpass the interactionlimit and exploit the full potential of ensemble quantum sensors. Insome embodiments, higher sensitivity and spin density can be achievedwithout being limited by the interaction limit.

In some embodiments of the present disclosure, the sensitivity and/orprecision of a dense ensemble of sensors, such as interacting electronicspins in diamond, can be improved beyond the sensitivity at the criticaldensity. For example, in some embodiments, a pulse sequence can bedesigned according to the disclosed design protocol such that, whenapplied to quantum sensors, decouples (reduces the effects to) thesensors from interactions and disorder. In some embodiments, such apulse sequence can also be fault-tolerant to the leading-order controlimperfections of the system. In some embodiments, pulse sequencesdesigned according to the disclosed protocol can enable a five-foldenhancement in coherence time (which refers to the time during which aquantum state remains stable, where longer coherence times expose thequantum sensors to a field to be measured for a longer time, therebyincreasing sensitivity thereto) compared to the conventional XY-8sequence. In some embodiments, the effective field experienced by thedriven sensors can be tailored by pulse sequences designed according tothe disclosed protocol and/or the disclosed sensor initialization andreadout protocol can be used to achieve high magnetic-field sensitivity.Example results from non-limiting applications of the disclosedtechniques demonstrate a 30% enhancement in sensitivity relative to theXY-8 sequence, breaking the sensitivity limit set by inter-sensorinteractions.

In some embodiments, the disclosed protocol is based on robust periodicmanipulation (Floquet engineering) of ensemble spin dynamics. Pulsesequences developed with the disclosed protocol deliver a sequence withhigh sensitivity to an external signal of interest, while simultaneouslydecoupling the effects of interactions and disorder, and remainingfault-tolerant against the leading-order effects of finite pulse widthand control imperfections. In some non-limiting example applicationsusing a dense ensemble of nitrogen-vacancy (NV) centers in diamond, afive-fold enhancement of the spin coherence time has been observedcompared to the conventional XY-8 sensing sequence that does notdecouple interactions. The disclosed design protocol can be used todesign a sequence such that the driven spins experience a maximaleffective sensing field in a particular direction (e.g., the[1,1,1]-direction). The spins can be prepared in a plane orthogonal tothe effective field to be measured to attain increased sensitivity. Thecombination of extended coherence time produced by example pulsesequences designed using the disclosed protocol and optimal sensing haveshown an example, non-limiting 30% enhancement in absolute sensitivityover the XY-8 sequence, achieving a volume-normalized sensitivity of28(1) nT·μm^(3/2)/√{square root over (Hz)}, among the best to date inthe solid state sensor systems.

Embodiments of the instant application relates generally to design ofpulse sequences for interacting spin ensembles. For example, thedisclosed embodiments provide a formalism that enables a design protocolfor the pulse sequence. The design protocol includes one or more designconditions for one or more pulses in the pulse sequence. The designconditions suppress undesired effects in the spin ensembles. Forexample, design conditions can be imposed to suppress disorder of thespin ensembles and/or interaction effects within the spin ensembles.Using the described formalism and design protocol, described herein arealso specific pulse sequences meeting design protocols described hereinto achieve these benefits. Additional design conditions can be imposedaccording to some embodiments to account for or suppress other effects,such as finite frame widths, interaction cross-terms, rotation angleerrors, or higher order effects.

For example, as described in further detail below, each frame of a pulsesequence includes an electromagnetic pulse that acts on the spin system.The pulses, for example, can be a sequence of π pulses and/or π/2pulses. Each pulse causes a positive or negative rotation of spinorientations along x, y, or z axes. The design protocol includes designconditions regarding one or more pulses in the pulse sequence. Forexample, in some embodiments, a design condition is that the numberpositive rotations along each axis should be the same as the numbernegative rotations along the respective axis. This design conditionsuppresses disorder in the spin ensemble. In addition, for example, someembodiments, a design condition is that the number of rotations alongone axis should be the same as the number of rotations along the otheraxes. This design condition suppresses interaction effects in the spinensemble. Further exemplary design conditions of the design protocol ofsome embodiments are described in further detail, below.

In some embodiments, the disclosed protocol can be based on adescription of the ensemble spin dynamics and average Hamiltonian interms of time-domain transformations of local Pauli spin operators. Thedisclosed simple algebraic conditions imposed on the transformations canprovide a systematic way to describe the engineering of interactions anddisorder, as well as the cancellation of dominant system imperfections,including errors in spin rotation as well as undesirable disorder andinteraction effects during the finite pulse durations. Moreover,embodiments of the disclosed protocol provide a recipe to developsystem-specific pulse sequences adapted to relevant parameters andtimescales in the Hamiltonian, and is also applicable to a wide range ofinteraction Hamiltonians beyond the conventional dipolar interaction.This enables the fault-tolerant implementation of target Hamiltoniansusing the disclosed protocol, enabling a variety of applications.

In some embodiments, an interaction-decoupling sequences can be tailoredto system characteristics and dominant energy scales in order to protectquantum coherence. In some additional or alternative embodiments, spindynamics can be engineered in the presence of an external AC targetmagnetic field into an effective evolution under a vectorial DC magneticfield as seen by the driven spins, enabling optimal quantum sensing inthe presence of interactions. In some additional or alternativeembodiments, the disclosed protocol can be used to engineer Hamiltonianswith a range of thermalization properties for quantum simulation.

Quantum sensors, such as nitrogen vacancy centers in diamond, can takeadvantage of quantum mechanical interactions to achieve superior spatialresolution, spectral resolution and sensitivity as opposed to classicalsensors. Such sensors can be used in a number of applications including,but not limited to serving as a powerful tool for the exploration offundamental physics and for applications in material science andbiochemical analysis. Other quantum sensors would benefit form the pulsesequences described herein, as would be apparent to one of skill in theart.

FIG. 1A is an illustration showing a spin ensemble quantum sensor,according to some embodiments. The spin ensemble quantum sensor includesa plurality of quantum spins, such as nitrogen vacancy (NV) centers 102,that together form an ensemble quantum sensor. Although the presentdisclosure describes quantum sensors embodied as NV centers, other typesof quantum sensors are contemplated, and the examples listed herein arenot limited to NV centers. For example, the disclosed design isgenerally applicable to many different spin systems, ranging frominteracting electronic spin ensembles, such as NV centers in diamond,phosphorus donors in silicon and rare earth ions, to conventional NMRsystems, and even to platforms of cold molecules. These differentsystems have a variety of competing energy scales that dominate thedynamics, which can be addressed with the use of the disclosed designprotocol to address the dominant effects. In some embodiments, a denseensemble of NV centers are used, such as an ensemble having an NV centerdensity of 45 ppm. The NV centers 102 can be contained within a material100, such as a in a black diamond beam having nanometer dimensions(“nanobeam”). Microwave 108 and optical excitation 104 are delivered tothe spins to control and read out their spin states using at least onemicrowave source 118 at least one laser 114, respectively. In someembodiments, the at least one laser 114 can produce optical excitation114 at different wavelengths. An external magnetic field 106, such as anAC magnetic field, can be used as a target sensing signal to be sensedby the NV centers 102. Fluorescence 112 from NV centers 102 can bedetected by an optical detector 120.

In some embodiments, the NV centers 102 can exhibit long-range magneticdipolar interactions between the spins as well as strong on-sitedisorder originating from paramagnetic impurities and inhomogeneousstrain in the diamond lattice. The bulk diamond 100 can be etched into ananobeam to improve control homogeneity and to confine the probingvolume to V=0.018 μm³. Other spin systems are contemplated.

FIG. 1B shows three magnetic sublevels, |0

and |±1

, in the ground state of example spins, such as NV centers, according tosome embodiments. In some embodiments, two levels, |0, −1) can beaddressed using resonant microwave driving using, for example, themicrowave source 118. In some embodiments, excitation from the externalmagnetic field 106 can cause a transition from the level |−1

to |+1

. The NV center 102 can then transition back to the level |−1

, by emitting fluorescence 120, which can be detected by an opticaldetector 112. In some embodiments, measurements can be performed at roomtemperature under ambient conditions.

In some embodiments, each NV center 102 ground state can be anelectronic S=1 spin. A static magnetic field can be applied to isolatean ensemble of effective two-level systems formed of NV centers with thesame crystallographic orientation. The spin states can be initializedand detected optically, and resonant microwave excitation can be used todrive coherent spin dynamics. An external AC magnetic field can also beused as a target sensing signal.

FIG. 2 shows the volume-normalized magnetic field sensitivity (η_(v)) asa function of spin density, ρ, according to some embodiments. The dashedline denotes the standard quantum limit scaling and solid curves showthe behavior when interactions between spins are taken into account fordifferent readout efficiency factors C. Beyond a critical density theenhancement in sensitivity plateaus due to a reduction in coherence timefor all values of C.

General Formalism

According to some embodiments, pulse sequences are described in terms ofinteracting picture spin frame orientations. Instead of thinking of aperiodic driving sequence in terms of the control pulses applied, theaverage Hamiltonian evolution is described using toggling spin frames,which correspond to going into the interacting picture with respect tocontrol pulses. Without being bound by theory, describing pulsesequences as such provides a simple formalism for describing thedecoupling performance for pulses meeting the design protocol describedherein. In addition, it also allows the formulation of concise criteriafor pulse imperfections described herein.

Without being bound by theory, in some embodiments, the design protocoland associated design conditions described herein is based on averageHamiltonian theory. This theory provides a framework to analyze thedynamics of an interacting spin ensemble under periodic manipulations.

In some embodiments, an interacting spin ensemble has a Hamiltoniangiven by the following relationH=H _(f) +H _(c)(t),  (1)where H_(f) is the free evolution Hamiltonian of the internal dynamicsof the spin ensemble, involving on-site disorder and spin-spininteractions, and H_(c)(t) describes the periodic control field. Asdescribed herein, the periodic control field includes an externalstimulus such as a pulse sequence (for example, of electromagneticpulses) that is applied to the spin ensemble.

In some embodiments, a sequence of periodic control pulses are appliedto the spin ensemble. For example, a sequence of periodic control pulse,such as that shown in FIG. 3 view (c), is applied to a spin ensemblewith a total period τ. In some embodiments, the unitary transformationinduced by the pulses is written asμ_(c)(t)=Σ exp[−i∫ ₀ ^(t) H _(c)(t ₁)dt ₁],  (2)where τ denotes time-ordering as shown in FIG. 3 view (c). In theinteracting picture representation with respect to the control fieldunitary U_(c)(t), the full evolution in one period of the sequence ofperiodic pulses is given byU(T)=τ exp[−i∫ ₀ ^(T) {tilde over (H)} _(f)(t ₁)dt ₁],  (3)where {tilde over (H)}_(f)(t₁)=U_(c) ^(†)(t₁)H_(f)U_(c)(t₁) is theinteracting picture Hamiltonian of the system, also known as thetoggling frame Hamiltonian. In some embodiments, the unitary evolutionoperator can be expanded using the Magnus expansion as follows:

$\begin{matrix}{{{U(T)} = {\exp\left\lbrack {- {{iT}\left( {H^{(0)} + H^{(1)} + \ldots} \right)}} \right\rbrack}},{where}} & (4) \\{{H^{(0)} = {\frac{1}{T}{\int_{0}^{T}{{{\overset{\sim}{H}}_{f}\left( t_{1} \right)}{dt}_{1}}}}}\ } & (5) \\{H^{(1)} = {{- \frac{i}{2T}}{\int_{0}^{T}{{dt}_{2}{\int_{0}^{t_{2}}{{{dt}_{1}\left\lbrack {{{\overset{\sim}{H}}_{f}\left( t_{2} \right)},{{\overset{\sim}{H}}_{f}\left( t_{1} \right)}} \right\rbrack}.}}}}}} & (6)\end{matrix}$

In certain embodiments, the above series is truncated to the zerothorder, for example, as is consistent with Average Hamiltonian theory.Without being bound by theory, this is a good approximation in theregime of fast periodic driving, where each Hamiltonian term iscancelled at a rate faster than the characteristic timescale of itscorresponding dynamics.

In some embodiments, the pulse sequence is a discrete pulse sequence asillustrated in FIG. 3 view (c). The integral for the average HamiltonianH _(ave) in Equation (5) of the discrete pulse sequence can be rewrittenas a summation over different evolution periods shown in FIG. 3 view (c)

$\begin{matrix}{{{\overset{\_}{H}}_{ave} = {\frac{1}{T}\left( {{\Sigma_{k}\tau_{k}H_{k}} + {\Sigma_{k}t_{p}H_{k,{k + 1}}^{\prime}}} \right)}},} & (7)\end{matrix}$where τ_(k), H_(k) are the duration and toggling frame Hamiltonian{tilde over (H)}_(f)(t) in the kth free evolution period, respectively,t_(p) is the finite pulse duration, and H′_(k,k+1) is the averageHamiltonian corresponding to the pulse between the kth and k+1th freeevolution period.

The above discussion is fully general, and may be applied to systemswith arbitrary spins. Some embodiments focus on an effective two-levelsystem, which can appear either in spin-½ systems, or as two levelsoff-resonant from other transitions in a higher spin system. In someembodiments, the form of the Hamiltonian is not restricted to the spin-½dipolar interaction Hamiltonian with on-site disorder, and instead usesgeneric Hamiltonians under the secular approximation, such that the onlyconstraint is that the Hamiltonian commutes with a strong magnetic fieldthat sets the quantization axis in the 2-direction, as described furtherherein. Some embodiments involve Hamiltonians with only single-body andtwo-body terms, although other embodiments involve interactions withthree-body operators. In this case, the Hamiltonian can generically bewritten asH ₀ =H _(dis) +H _(int)=Σ_(i)Δ_(i) S _(i) ^(z)+Σ_(i,j)[J _(ij) ^(ex,r)(S_(i) ^(x) S _(j) ^(x) +S _(i) ^(y) S _(j) ^(y))+J _(ij) ^(ex,i)(S _(i)^(x) S _(j) ^(y) −S _(i) ^(y) S _(j) S ^(x))+J _(ij) ^(is) S _(i) ^(z) S_(j) ^(z)],  (8)where the first term H_(dis) is the disorder Hamiltonian and the secondterm H_(int) is a generic interaction Hamiltonian, Δ_(i) is a randomon-site disorder strength, S_(i,j) are spin-½ operators, and J_(ij)^(ex,r), J_(ij) ^(ex,i), J_(ij) ^(is) are arbitrary interactionstrengths for the spin-exchange interaction with real and imaginaryhopping phase and the Ising interaction. The real spin-exchangeinteraction and Ising interaction are widely found in differentinteracting spin systems, and the imaginary spin-exchange interactioncan also emerge in experimental systems of certain embodiments. In someembodiments, it is assumed that the periodic control field satisfiesH _(c)(t)=Σ_(i)(i+δ _(i))(Ω_(x)(t)S _(i) ^(x)+Ω_(y)(t)S _(i) ^(y)),  (9)where δ_(i) describes the control field strength inhomogeneity andΩ_(x)(t), Ω_(y)(t) are the instantaneous Rabi frequencies. In someembodiments it is assumed that the pulses are applied along the {rightarrow over (x)} or {right arrow over (y)} directions only, with arotation angle that is an integer multiple of π/2. However, thoseskilled in the art will recognize that the formalism and design protocolcan also be applied to other pulses without these assumptions.Frame Representation

Described herein is a frame representation that is used to describedesign conditions of the design protocol for a pulse sequence acting on,for example, a spin ensemble. In some embodiments, a pulse sequenceincludes a series of frames. Each frame includes one or more pulses, orpulses that can be described as one or more pulses.

In some embodiments, each frame can be represented based on theinteracting picture transformations of the S^(z) operator: {tilde over(S)}^(z)(t)=U_(c) ^(†)(t)S^(z)U_(c)(t)=Σ_(μ=x,y,z)F_(μ)(t)S^(μ),hereinafter referred to as the toggling frame spin operator. Thefunctions F_(μ)(t) are vector generalizations of the modulation functionemployed in the analysis of sensing sequences. Without being bound bytheory, in the presence of a strong field that sets the quantizationaxis in the 2 direction and under the secular approximation, accordingto certain embodiments, the Hamiltonian described in equation (8) isinvariant under rotations around the z axis that mix S^(x) and S^(y).Under such conditions, the Hamiltonian does not depend independently ontransverse components, and information about {tilde over (S)}^(z)(t) issufficient to fully recover the interacting picture Hamiltonian.

An example of this representation is illustrated for various pulsesequences, such as decoupling pulses, in FIG. 4. Pulses in the sequenceare applied as π and π/2 pulses, as shown in the top parts of FIG. 4views (a)-(c). These pulses lead to rotations of the spin operators onthe Bloch sphere as shown in the bottom parts of FIG. 4 views (a)-(c).The pulses can be viewed in an alternative representation, as shown inthe top part of FIG. 4 views (d) to (f). Each cell represents a freeevolution period before or after the pulses above in FIG. 4 views (a) to(c). The grey cells represent positive rotations in the respectiveF_(x), F_(y), and F_(z) direction, whereas the black cells representnegative rotations in the respective direction. The bottom part of FIG.4 views (d) to (f) shows the effects of each sequence on the system. Thefirst circle in the bottom part of FIG. 4 views (d) to (f) representswhether the sequence suppresses disorder, with a check indicating thatdisorder is suppressed and an “X” indicating no suppression. The secondcircle in the bottom part of FIG. 4 views (d) to (f) represents whetherthe sequence suppresses interaction effects, with a check indicatingthat interaction effects are suppressed and an “X” indicating nosuppression.

The first pulse sequence, which corresponds to FIG. 4 views (a) and (d),represents a spin echo pulse sequence. Such a pulse sequence can cancelthe effects of on-site disorder, but leaves interactions unchanged. FIG.4 views (b) and (e) represents a WAHUHA pulse sequence, which suppressesdipolar interactions, but leaves on-site disorder intact. FIG. 4 views(c) and (f) represents a sequence according to some embodiments. Thesequence of FIG. 4 views (c) and (f) cancels both on-site disorder anddipolar interactions to leading order in the ideal pulse limit.

The top parts of FIG. 4 views (a) to (c) represent the pulses that areapplied in the sequence. According to some embodiments, the pulses areeither π pulses or π/2 pulses. Positive pulses are shown above the axisand negative pulses are shown below the axis. π pulses are shown thickerthan π/2 pulses.

The bottom parts of FIG. 4 views (a) to (c) represent the evolution ofthe toggling frame spin operators after each pulsed rotation in thesequence. In this representation, the pulses specify rotations around anexternal axis, i.e. an {circumflex over (x)} rotation is applied aroundthe {circumflex over (x)} axis in a fixed external reference frame, asopposed to rotating around the modified 2-arrow direction. The arrowsthat point in the {circumflex over (x)}, ŷ, {circumflex over (z)}directions in the external frame then correspond to the interactingpicture {tilde over (S)}^(x)(t), {tilde over (S)}^(y)(t), {tilde over(S)}^(z)(t) operators at each time. For example, in the second Blochsphere in FIG. 4 view (b), the ŷ arrow is pointing in the {circumflexover (z)} direction, indicating that {tilde over (S)}_(z)(t)=S^(y)during this free evolution period.

FIG. 4 views (d) to (f) are further representations of the rotationscaused by the pulses in the pulse sequences of FIG. 4 views (a) to (c).In these views, the location of the filled square in each columnindicates the axis direction that {tilde over (S)}^(z)(t) points in, andthe color indicates a positive (gray) or negative (black) directionalong this axis. These cells in these views correspond to the Blochspheres shown in the bottom part of FIG. 4 views (a) to (c).

The representations in FIG. 4 views (d) to (f) can be expressedmathematically as a matrix F, with different rows characterizing theoperator pointing during each free evolution block, and ±1 indicate thatthe operator is pointing in the positive or negative direction in eachblock. For instance, for the example illustrated in FIG. 4 view (f) canbe mathematically expressed as:

$\begin{matrix}{F = {\begin{pmatrix}0 & 0 & {+ 1} & {- 1} & 0 & 0 \\0 & {+ 1} & 0 & 0 & {- 1} & 0 \\{+ 1} & 0 & 0 & 0 & 0 & {- 1}\end{pmatrix}.}} & (10)\end{matrix}$

In the following, it is assumed that free evolution times are an integermultiple of a base duration. However, the representation above anddesign protocol of design conditions according to this disclosure is notlimited thereto, and is readily generalizable to arbitrary pulseseparations, for example, by additionally specifying the duration spentin each free evolution block, as would be understood by one of ordinaryskill in the art.

Exemplary Decoupling Design Conditions

As described in this section, in some embodiments, pulse sequences adesign protocol can be constructed with design conditions based on idealpulses, i.e., without considering pulse imperfections. Embodimentsdealing with pulse imperfections are described in further sections.

Using the frame representation described herein, each S^(z) operator canbe described vt Σ_(μ=x,y,z)F_(μk)S^(μ) in the Hamiltonian for the k^(th)free evolution block, where F_(μk) denotes the (μ,k) element of thematrix F. Using this, each of the average Hamiltonian terms in the idealpulse limit can be determined. In some embodiments, it is assumed thateach free evolution block has equal duration τ. With this assumption,the disorder Hamiltonian can be expressed as

$\begin{matrix}{{{\overset{\_}{H}}_{dis} = {\frac{1}{N}{\sum\limits_{i,k,\mu}{\Delta_{i}F_{\mu\; k}S_{i}^{\mu}}}}},} & (11)\end{matrix}$while the three types of interaction terms can be respectively expressedas

$\begin{matrix}{{{\overset{\_}{H}}_{{ex},r} = {\frac{1}{N}{\sum\limits_{i,j,k}{J_{ij}^{{ex},r}{\sum\limits_{\mu}{\left( {1 - F_{uk}^{2}} \right)S_{i}^{\mu}S_{j}^{\mu}}}}}}},} & (12) \\{{{\overset{\_}{H}}_{{ex},i} = {\frac{1}{N}{\sum\limits_{i,j,k}{J_{ij}^{{ex},i}{\sum\limits_{\mu,v,\sigma}{\epsilon_{\mu\; v\;\sigma}{F_{\mu\; k}\left( {{S_{i}^{v}S_{j}^{\sigma}} - {S_{i}^{\sigma}S_{j}^{v}}} \right)}}}}}}},} & (13) \\{{{\overset{\_}{H}}_{is} = {\frac{1}{N}{\sum\limits_{i,j,k}{J_{ij}^{is}{\sum\limits_{\mu}{F_{\mu\; k}^{2}S_{i}^{\mu}S_{j}^{\mu}}}}}}},} & (14)\end{matrix}$where N is the total number of free evolution blocks in the sequence and∈_(μvσ) is the Levi-Civita symbol.

There are two types of functional dependencies on F_(μk) in Equations(11)-(14): the disorder Hamiltonian H _(dis) and imaginary spin-exchangeHamiltonian Ĥ_(ex,i) involve terms linear in F_(μk), while the realspin-exchange Hamiltonian H _(ex,r) and Ising Hamiltonian H _(is)involve terms quadratic in F_(μk). Since each element F_(μk) takes onvalues 0, ±1, the quadratic term can also be written as F_(μk)²=|F_(μk)|.

In some embodiments, the disorder can be reduced by setting a designcondition of the design protocol to require that the positive rotationsare the same as the number of negative rotations in each of x, y, and z.For example, based on the linear dependence on F_(μk) of H _(dis) and H_(ex.i), these average Hamiltonian contributions can be cancelled wheneach row μ has an equal number of positive and negative elements, suchthat they sum to 0. Physically, in some embodiments, this corresponds toguaranteeing, in which each disorder precession around a positive axisis compensated by one in the opposite direction. Applying this criteriato the sequences in FIG. 4, the spin echo represented in FIG. 4 views(a) and (d) and sequence of FIG. 4 views (c) and (f) cancel disorder,while the WAHUHA sequence does not (as shown in FIG. 4 views (b) and(e)). That is, in each of these sequences, for every positive rotationin x, y, or z in the spin echo and the sequence in FIG. 4 views (c) and(f) there is also a negative rotation in x, y, or z, whereas this is notthe case in the WAHUHA sequence.

In some embodiments, interaction effects can be suppressed by setting adesign condition of the design protocol to require that the rotations ineach of x, y, and z are the same. This can be described as symmetrizinginteractions. Despite the quadratic dependence on F_(μk) of H _(ex,4)and H _(is) the interactions can be symmetrized into a HeisenbergHamiltonian by choosing the sum Σ_(k)|F_(μk)| to be the same for each μ.In some embodiments, this preserves coherence, for example, becauseglobally polarized initial states that can be prepared experimentallyconstitute an eigenstate of the Heisenberg interaction, and consequentlythere will be no dephasing under the Heisenberg Hamiltonian. Therefore,the spin echo represented in FIG. 4 views (a) and (d) does notsymmetrize interactions, while the WAHUHA sequence as represented inFIG. 4 views (b) and (e) and the sequence of FIG. 4 views (c) and (f)symmetrize the interaction Hamiltonian into a Heisenberg form.

According to the above, according to some embodiments, a design protocolcan be expressed with the following two design conditions on the matrixF for dynamical decoupling (disorder and interactions) as:

Disorder H _(dis) and imaginary spin-exchange H _(ex,i) suppression:each row sums to 0, i.e.

$\begin{matrix}{{{\sum\limits_{k}F_{\mu\; k}} = 0},{{{for}\mspace{14mu}\mu} = 1},2,3.} & (15)\end{matrix}$Ising interaction H _(is) and real spin-exchange H _(ex,r) symmetrizedinto Heisenberg form: each row should have an equal number of nonzeroelements, i.e.

$\begin{matrix}{{\sum\limits_{k}{F_{1k}}} = {{\sum\limits_{k}{F_{2k}}} = {\sum\limits_{k}{{F_{3k}}.}}}} & (16)\end{matrix}$These design conditions can be extended to the case where different freeevolution blocks have different length by weighting each of the terms bytheir corresponding evolution time. In addition, while the above focuseson single-body and two-body interactions, the design conditions can beextended to an interaction involving more spins, as described furtherherein. For example, as described herein, in some embodiments resultsfrom unitary t-designs prove that the same design conditions describedabove also guarantee decoupling three-body interaction effects forpolarized initial states.Decoupling Design Conditions for Finite Pulse Imperfections

According to some embodiments, design protocols can also include designconditions that take into account pulse imperfections in the pulsesequences. The formalism and frame representation described herein canalso be used to express such design conditions that take into accountaverage Hamiltonian finite pulse imperfections. For example, thesecorrespond to the average Hamiltonian H′_(k,k+1) during the pulseintroduced in equation (7), and includes contributions from disorder andinteractions, interaction cross-terms, and/or rotation angle errors,according to some embodiments.

As described in further detail herein, some embodiments of the designprotocol can include design conditions to cancel the following pulseduration effects:

Disorder and Direct Interaction Contributions:

In some embodiments, all pulses can be rephrased as all pulses in termsof π/2 pulse building blocks, and intermediate frame orientationsbetween each π/2 pulse of a full π are accounted for (for example asdescribed with reference to FIG. 5A). For example, a π pulse can betreated as a concatenation of two π/2 pulses with no time separation, asshown in the top of FIG. 5A. The middle of FIG. 5A shows the Blochspheres that result from treating the π pulse as two π/2 pulses.Therefore, in some embodiments, the design protocol can take intoaccount an intermediate frame between the two π/2 pulses (as shown inthe bottom of FIG. 5A). Intermediate frames, such as these, can bereferred to as patching frames. These patching frames can be representedas a narrow bar between free evolution periods T, as shown for examplein FIGS. 5C-5D.

Interaction Cross Terms:

In some embodiments, for each pair of axes, the “parity” of framechanges sums to 0. Mathematically, this can be written as the condition

$\begin{matrix}{{{{\sum\limits_{k}{F_{\mu,k}F_{v,{k + 1}}}} + {F_{\mu,{k + 1}}F_{v,k}}} = 0},} & (17)\end{matrix}$for (μ, v)=(1, 2), (1, 3), (2, 3), where the summation is a cyclic sumover all evolution blocks. Physically, this expression characterizeswhether the operator orientation changed sign.

Rotation Angle Errors:

In some embodiments, for each pair of axes, the “chirality” of framechanges sums to 0. Mathematically, this can be written as the condition

$\begin{matrix}{{{\sum\limits_{k}{{\overset{\rightarrow}{F}}_{k} \times {\overset{\rightarrow}{F}}_{k + 1}}} = \overset{\rightarrow}{0}},} & (18)\end{matrix}$where {right arrow over (F)}_(k)=Σ_(μ)F_(μ,k){right arrow over (e)}_(μ),{right arrow over (e)}_(μ) are the unit vectors along the axisdirections, and the summation over k is a cyclic sum over all evolutionblocks. Physically, this expression is a cross product betweenneighboring spin operator orientations, which corresponds to thedirection of the rotation axis in the toggling frame.

While some embodiments described herein focus on single-body andtwo-body terms, one of ordinary skill would understand that thedescription can be generalized to interactions involving three spins.

In some embodiments, frames during the free evolution period alsoprovide a description of imperfections during the driven evolution. Insome embodiments, additional effects such as waveform transients andpulse shape imperfections are analyzed in a similar fashion to leadingorder, although the pulses are assumed to be Markovian (i.e. the unitaryoperator of one pulse is not modified by its preceding pulses).

Finite Pulse Widths

In some embodiments, the design protocol includes design conditions thattake into account the effects of pulse widths, i.e., finite pulsedurations, that take place between the free evolution periods

. In some embodiments, the effects of a finite pulse duration t_(p) canbe determined by calculating the average Hamiltonian, for example, for aπ/2 pulse building block. For illustration the average Hamiltonianduring π/2 pulse that causes a rotation of the frame from +z into +y canbe described as

$\begin{matrix}{{{\overset{\_}{H}}_{\pi\text{/}2} = {\frac{2}{\pi}{\int_{0}^{\pi}{U_{0}^{\dagger}{U(\theta)}^{\dagger}H_{f}{U(\theta)}U_{0}d\;\theta}}}},} & (19)\end{matrix}$where U₀ is the unitary evolution operator due to the preceding pulses,and U(θ) describes the rotation of the π/2 pulse up to angle θ.Physically, this corresponds to the S^(z) operator being continuouslytransformed as S^(z)(θ)=S^(z) cos θ+S^(y) sin θ in the toggling frame(FIG. 5B). Carrying out the integral over the angle, the averageHamiltonian during the π/2 pulse can be rewritten as

$\begin{matrix}{{\overset{\_}{H}}_{\pi\text{/}2} = {{\frac{2}{\pi}\left( {H_{1}^{dis} + H_{2}^{dis}} \right)} + {\frac{2}{\pi}\left( {H_{1}^{{ex},i} + H_{2}^{{ex},i}} \right)} + {\frac{1}{2}\left( {H_{1}^{is} + H_{2}^{is}} \right)} + {\frac{1}{2}\left( {H_{1}^{{ex},r} + H_{2}^{{ex},r}} \right)} + {\sum\limits_{i,j}{\frac{J_{ij}^{is} - J_{ij}^{{ex},r}}{\pi}{\left( {{S_{i}^{Z}S_{j}^{y}} + {S_{i}^{y}S_{j}^{z}}} \right).}}}}} & (20)\end{matrix}$Thus, the average Hamiltonian during a pulse of finite duration is givenby the average of the Hamiltonians in the initial and final frames(i.e., ½(H₁ ^(is)+H₂ ^(is))+½(H₁ ^(ex,r)+H₂ ^(ex,r)), and as shown inFIG. 5B), plus a cross-term

$\left( {{i.e.},{\Sigma_{i,j}\frac{J_{ij}^{is} - J_{ij}^{{ex},r}}{\pi}\left( {{S_{i}^{Z}S_{j}^{y}} + {S_{i}^{y}S_{j}^{z}}} \right)},} \right.$relating to the rotation of the frame from +z to +y). As would beapparent to one of ordinary skill in the art, similar expressions willalso hold for other π/2 rotations other than a frame change from +z to+y, such as rotations from +z to +x, +y to +z, +y to +x, +x to +z, and+x to +y. In addition, in some embodiments, this analysis can be appliedto other pulses using a π/2 pulse as a building block. For example, insome embodiments this analysis can be applied to π pulses, which can beregarded as a combination of two π/2 pulses with the specification ofthe patching frame.

Therefore, in some embodiments, each free evolution period of duration

as an effective free evolution of duration

+4t_(p)/π for disorder and imaginary spin-exchange (as shown in FIG.5C), and as duration

+t_(p) for real spin-exchange and Ising interactions (as shown in FIG.5D). Therefore, in some embodiments, the design conditions for disorderand interaction decoupling described in equations (15) and (16) canfurther take into account the average terms in the patching frames(intermediary frames) preceding and following each free evolution period

that result from finite pulse width, as shown in FIGS. 5C-5D. Forexample, as shown in FIG. 5C, there is one positive and one negativeh_(i)S_(i) ^(y) patching frame for F_(y), satisfying the designcondition imposed by equation (15). Applying this design condition tothe patching frames resulting from finite pulse duration suppressesdisorder. In addition, for example, as shown in FIG. 5D, there is onefinite pulse interaction patching frame for each of F_(x), F_(y), andF_(z), satisfying the design condition imposed by equation (16).Applying this design condition to the patching frames resulting fromfinite pulse duration suppresses interaction effects.

In some embodiments, design conditions can also be included to addressthe cross term

$\left( {\Sigma_{i,j}\frac{J_{ij}^{is} - J_{ij}^{{ex},r}}{\pi}\left( {{S_{i}^{Z}S_{j}^{y}} + {S_{i}^{y}S_{j}^{z}}} \right)} \right)$described above. Since the sign of the interaction cross-termcorresponds to the product of the parity of initial and final frames,for each pair of directions, a design condition according to someembodiments requires that the number of neighboring frames with apositive parity product be equal to the number of neighboring frameswith a negative parity product (for example as described in equation(17) and illustrated in FIG. 5E).

The frame representation discussed above with respect to the freeevolution period can be extended to systematically analyze pulseduration imperfections. In some embodiments, π/2 pulses are used as thebasic building block, all other types of pulses are phrased in terms ofthem. In some embodiments, this representation has the advantage thatimperfections become much easier to analyze, as discussed furtherherein, and that each rotation can be uniquely specified by the frameorientations.

Using π/2 pulses as the basic building block, according to someembodiments, requires the additional specification of “patching” framesfor pulses of larger rotation angles. These patching frames keep trackof the intermediate spin, for example the spin pointing at the middle ofa it pulse. This corresponds to decomposing the it pulse into two π/2pulses, one right after another (for example as shown in FIG. 5B). Insome embodiments, an alternative way to view a it-pulse is a sequence ofπ/2 pulses with a free evolution period of zero time in between the twoπ/2 pulses. Therefore, the frame representation can be extended toinclude these short patching frames by adding an additional row toindicate the duration of free evolution for each frame orientation. Forexample, the spin echo pulse sequence of FIG. 4 views (a) and (d) can bewritten as

$\begin{matrix}{{F = \begin{pmatrix}0 & 0 & 0 \\0 & {+ 1} & 0 \\{+ 1} & 0 & {- 1} \\\tau & 0 & \tau\end{pmatrix}},} & (21)\end{matrix}$where the middle column is the patching frame introduced for the πpulse. The patching frame has duration zero, as shown in the bottom row,whereas the two π/2 pulses have duration τ. Similarly, the pulsesequence of FIG. 4 views (c) and (f) and equation (10) can be written as

$\begin{matrix}{F = {\begin{pmatrix}0 & 0 & {+ 1} & 0 & {- 1} & 0 & 0 \\0 & {+ 1} & 0 & {+ 1} & 0 & {- 1} & 0 \\{+ 1} & 0 & 0 & 0 & 0 & 0 & {- 1} \\\tau & \tau & \tau & 0 & \tau & \tau & \tau\end{pmatrix}.}} & (22)\end{matrix}$

In some embodiments, a patching frame can also be introduced for π/2pulses, in which the usual π/2 rotation is implemented by a compositepair of π/2 pulses along different axes, instead of by a single pulse.These patching frames for π/2 pulses can be important, for example, inmaximizing sensitivity and maintaining efficient dynamical decoupling asdiscussed further herein.

Rotation Angle Errors

In some embodiments, the design protocol can also include designconditions that take into account control errors, such as systematicrotation angle error. For example, a rotation angle error around the+{circumflex over (x)} direction can be compensated with anotherrotation around the −{circumflex over (x)} direction. In someembodiments, rotation angle error effects can be analyzed directly forthe average Hamiltonian. This can be different from the rotation axisapplied in the lab frame in some embodiments.

For example, for a π/2-pulse

$P_{k} = {\exp\left( {{- i}\frac{\pi}{2}A_{k}} \right)}$around some axis A_(k), according to some embodiments, a rotation angleerror corresponds to an actual rotation

${\overset{˜}{P}}_{k} = {\exp\left\lbrack {{- {i\left( {\frac{\pi}{2} + {\delta\theta}} \right)}}A_{k}} \right\rbrack}$being applied. In some embodiments, this can be equivalently regarded asan error term

$\frac{\delta\theta}{t_{p}}A_{k}$acting during the rotation. The average Hamiltonian in the togglingframe corresponding to this can be calculated as H _(r,k)=U_(l−1)^(\)H_(r,k)U_(k−1). This can be rewritten by making use of theexpressions for the initial and final frames of this pulse:

$\begin{matrix}{{{U_{k - 1}^{\dagger}S^{z}U_{k - 1}} = {\sum\limits_{\mu}{F_{u,{k - 1}}S^{\mu}}}},} & (23) \\{{U_{k - 1}^{\dagger}P_{k}^{\dagger}S^{z}P_{k}U_{k - 1}} = {{U_{k}^{\dagger}S^{z}U_{k}} = {\sum\limits_{\mu}{F_{u,k}{S^{\mu}.}}}}} & (24)\end{matrix}$Inserting U_(k−1)U_(k−1) ^(\)=I into the latter equation gives thefollowing expression

$\begin{matrix}{{{\left( {U_{k - 1}^{\dagger}P_{k}U_{k - 1}} \right)^{\dagger}\left( {\sum\limits_{v}{F_{v,{k - 1}}S^{v}}} \right)\left( {U_{k - 1}^{\dagger}P_{k}U_{k - 1}} \right)} = {\sum\limits_{\mu}{F_{u,k}S^{\mu}}}},} & (25)\end{matrix}$U_(k−1) ^(\)P_(k)U_(k−1) is the rotation pulse exp

$\left( {{- i}\frac{\pi}{2}B_{k}} \right)$that, in the toggling frame representation, rotates the k−1th frame intothe kth one. In some embodiments, the rotation axis direction B_(k) isobtained from a cross product between the initial and final togglingframe directions, which corresponds to each term in equation (18).

Since

${P_{k} = {\exp\left( {{- i}\frac{\pi}{2}A_{k}} \right)}},$where A_(k) is proportional to a Pauli matrix, the unitary U_(k−1) canbe moved into the exponential

$\begin{matrix}{{{U_{k - 1}^{\dagger}P_{i}U_{k - 1}} = {{\exp\left( {{- i}\frac{\pi}{2}U_{k - 1}^{\dagger}A_{k}U_{k - 1}} \right)} = {\exp\left( {{- i}\frac{\pi}{2}B_{k}} \right)}}}.} & (26)\end{matrix}$

Thus, the average Hamiltonian can be expressed as

$\begin{matrix}{{{\overset{\_}{H}}_{r,k} = {U_{k - 1}^{\dagger}\frac{\delta\;\theta}{t_{p}}B_{k}}},} & (27)\end{matrix}$which is uniquely specified by the initial and final frames of therotation pulse, as determined by the chirality of the frame changeEquation (18).

In some embodiments, an alternative geometric picture of the precedingderivation is as follows: the analysis starts with an internal framecoordinate system on the Bloch sphere (arrows) that coincides with theexternal reference coordinate system, as shown, for example, in thefirst Bloch sphere of the bottom panel of FIG. 4 view (c). Rotations areapplied along the axes of the fixed external coordinate system, whichrotate the toggling frame coordinate system. The direction of thetoggling frame coordinate system that coincides with the external +zdirection corresponds to {tilde over (S)}_(z)(t)=U_(c) ^(\)S^(z)U_(c).To specify the axis of rotation in the toggling frame however, it isconjugated by all of the preceding pulses (this corresponds to followinginto the toggling frame coordinate system, and determining which axisthe rotation is being applied around in this coordinate system). Sincethe initial and final frame directions in the toggling frame coordinatesystem are known, the rotation effect is uniquely specified and can beeasily characterized by the chirality of the rotation connecting theinitial and final frames.

Therefore, in some embodiments, cancellation of rotation angle errors iswell-captured by the chirality of each frame change. For example, asshown in FIG. 5F, systematic rotation angle errors are suppressed bychoosing opposite chirality between frame changes. Note that since theabove discussion considers single-body terms, the analysis applies toboth a uniform rotation angle error as well as cases with a fieldinhomogeneity, according to some embodiments. Rotation axis errors canalso be incorporated, as discussed further herein.

Exemplary Sequence Design

The embodiments discussed herein describe algebraic rules (designconditions) for the suppression of disorder, interactions, and one ormore kinds of finite pulse duration effects that together make up adesign protocol, according to some embodiments. The formalism and framerepresentation discussed above greatly simplifies the design procedureand enables versatile pulse sequences depending on the dominant effectsin the system. According to some embodiments, general statements aboutsequence optimality and the benefit of certain structures in the pulsesequence can be made.

For example, the minimal number of free evolution blocks required toachieve full suppression of certain effects can be constrained, in someembodiments. Without being bound by theory, in the case of ideal pulses,full cancellation of interactions, for instance, requires at least 3free evolution blocks in order to achieve the necessary symmetrization,while full cancellation of both interactions and disorder requires 6free evolution blocks to cover the ±{circumflex over (x)}, ±ŷ,±{circumflex over (z)} directions.

In some embodiments, to suppress all average Hamiltonian terms in thepresence of pulse imperfections, at least 12 free evolution blocks arerequired. This is because the parity condition equation (17) forinteraction cross terms requires one of the frame orientations to appeartwice with the same sign, so in order to achieve the symmetrizationbetween all directions, at least 12 free evolution blocks are required.One embodiment of such a pulse sequence, for example, can be representedas

$\begin{matrix}{{F = \begin{pmatrix}0 & 0 & 1 & 0 & 0 & {- 1} & 0 & 0 & {- 1} & 0 & 0 & 1 \\0 & 1 & 0 & 0 & {- 1} & 0 & 0 & 1 & 0 & 0 & {- 1} & 0 \\1 & 0 & 0 & {- 1} & 0 & 0 & {- 1} & 0 & 0 & 1 & 0 & 0\end{pmatrix}},} & (28)\end{matrix}$(with the fourth row time label not included as all pulses are of equalduration). According to some embodiments, this pulse sequence decouplesall average Hamiltonian terms with only 12 free evolution blocks.Suppressing Higher Order Effects

Although some embodiments described herein focus on average Hamiltoniantheory, higher-order effects can also be incorporated. For example, insome embodiments the Magnus expansion, as described in equations(4)-(6), can be utilized. Without being bound by theory, in this casethe average Hamiltonian theory corresponds to truncating the Magnusexpansion to zeroth order. However, some embodiments retain more ordersof the Magnus expansion and thereby higher-order effects can besuppressed.

First, in some embodiments, combining a pulse sequence with itstime-reversed counterpart automatically suppresses all first-orderMagnus expansion contributions. In some embodiments, such as withapplications such as AC sensing, design protocols are designed so thatthe time-reversed counterpart does not cancel the desired sensing fieldcontributions.

Second, in some embodiments, the magnitude of higher-order contributionscan be reduced by taking into account the dominant energy scales in thesystem and designing pulse sequences to suppress such effects first, asone of ordinary skill would understand in light of this disclosure.Without being bound by theory, this can be because the commutators ofother evolution pieces with the dominant Hamiltonian terms cancel. Insome embodiments, this type of higher-order suppression can be extendedto disorder-dominated or control-inhomogeneity-dominated systems, asdiscussed further herein. In some embodiments, the technique ofsecond-averaging in NMR can also allow engineering a dominant term inthe Hamiltonian, such that the effects of undesired contributions aresuppressed. In some embodiments, the fastest timescale of the pulsesequence is designed to suppress the highest priority effect orcontribution.

Third, in some embodiments, since the representation described hereincan provide a complete description of the Hamiltonian at all times, thedesign conditions can be rewritten to directly incorporate higher-orderMagnus terms. In some embodiments, this design process can be simplifiedby using techniques such as pulse cycle decoupling, which allowsanalysis of the higher-order decoupling properties of a sequence basedon its individual building blocks, as would be understood by one ofordinary skill in the art in light of this disclosure.

Example Pulse Sequences

FIG. 6 depicts pulsed decoupling sequences with robustness to differenttypes of dominant energy scales, according to certain embodiments. FIG.6 view (a) depicts a pulse sequence, Sequence A (Cory-48), according toone embodiment, where axis permutation is performed on the fastesttimescale, prioritizing the cancellation of interaction effects. FIG. 6view (b) depicts a pulse sequence, Sequence B, according to oneembodiment, where spin echoes are performed on the fastest timescale,prioritizing disorder decoupling. FIG. 6 views (a) and (b) also showframe representations for the first 12 free precession intervals of eachsequence, including the time-scales for interaction decoupling (T_(J))and disorder decoupling (T_(W)). FIG. 6 view (c) depicts T₂-decay timefor the sequences in (a) (outlined in black) and (b) (not outlined) as afunction of disorder and interaction strength, where a clear crossoverof performance is observed. Pulses are assumed to be infinitesimallyshort and τ=20 ns.

In some embodiments, the techniques described in the present disclosurecan be combined with other techniques, such as spin-bath engineering,photon-collection optimization, double-quantum magnetometry, and noveldiamond growth techniques, which together can push the volume-normalizedsensitivity even lower, for example below the picotesla level in a μm³volume for spin densities above 50 ppm. Such a sensitivity can beapplied, according to some embodiments, to many applications, such asnanoscale nuclear magnetic resonance and investigations of stronglycorrelated condensed matter systems. In some embodiments, examplefault-tolerant sequences designed according to the disclosed protocolcan be generate a broad class of many-body Hamiltonians, for examplebased on a time-domain transformations of local Pauli spin operators,providing a tool to apply non-equilibrium phenomena in driven quantumsystems as well as to create highly entangled states for, for example,interaction-assisted quantum metrology.

Example Design Protocol Effects

Without being bound by theory, example embodiments of the disclosedprotocol for designing pulse sequences go beyond the limitations ofcurrent design methods in the following aspects, as discussed throughoutthe present disclosure.

Fault-Tolerance:

Different types of average Hamiltonian effects, including those offinite pulse imperfections, can be readily incorporated as concise,intuitive algebraic conditions on the choice of toggling frameorientations, according to some embodiments. This provides, in someembodiments, a protocol to design pulse sequences for interacting spinensembles that have robustness automatically built in.

Generality of Hamiltonian:

Embodiments of the present disclosure can be used for general spin-½Hamiltonians under the secular approximation (e.g., rotating waveapproximation in a strong external magnetic field). This can include thestandard on-site disorder and dipolar interaction terms, and/or can alsoextend to two-body imaginary spin-exchange interactions, and/or genericinteractions involving up to three-body spin operators.

Versatile Adaptation to Different Experimental Systems:

Existing NMR approaches use pairs of π/2 pulses (solid echo pulse block)for interaction decoupling. This approach can decouple interactionsfirst, but may only be effective for systems dominated by dipolarinteractions. Embodiments of the disclosed protocol allow for the designof pulse sequences that are specifically tailored to other types ofsystems, where disorder or control inhomogeneities are dominant, evenwhile interactions still play a role in the dynamics (see FIG. 3, view(a)).

Optimality:

In some embodiments, using simple algebraic conditions describedthroughout the present disclosure for decoupling of various effects, theshortest length to achieve full decoupling of a given set of terms canbe determined. Embodiments of the disclosed combinatorial analysis canbe used to provide sequences that can provide optimal sensitivity to ACsignals.

Adaptable to Many Applications:

Embodiments of the disclosed protocol can be adapted to manyapplications beyond magnetometry using NV centers, such as quantumsensing and quantum simulation, which can impose additional requirementsthat can be input into the disclosed protocol.

Example Design Protocol Applications

Example applications of embodiments of the disclosed protocol include,but are not limited to:

Protecting Quantum Information:

In some embodiments, interacting spin ensembles can be dynamicallydecoupled to protect the coherence of an initial state. In some exampleapplications, disorder terms in the Hamiltonian can be fully canceled,and interactions can be symmetrized into a Heisenberg form. Since theexample polarized initial states are an eigenstate of the HeisenbergHamiltonian, such embodiments will not experience dephasing under suchan interaction, and the coherences will be long-lived, therebyprotecting quantum information stored in the initial state.

Quantum Sensing:

In addition to the conditions for dynamical decoupling describedthroughout the present disclosure, embodiments of sequences developedusing the disclosed protocol can have well-defined sensing resonances,such that efficient phase accumulation is incurred due to an externalfield to be sensed. Without being bound by theory, the Fourier transformof the disclosed toggling frame representation can demonstrate thisexample property. In some embodiments, optimizing sensitivity caninvolve aligning the resonance frequency and phase for each of the axesdirections to a field to be sensed, as well as or in the alternative tochoosing optimal initialization and readout directions describedthroughout the present disclosure.

Quantum Simulation:

In some embodiments, the conditions for full decoupling can be relaxed,allowing the disclosed protocol to engineer specific types of many-bodydisorder and interaction Hamiltonians. For example, the same descriptionfor the cancellation of certain Hamiltonian terms can be rephrased forthe engineering of the Hamiltonian, by changing the condition from beingequal to zero into being a finite value, for example as described in thedescription of Quantum Simulation below.

Disorder and Interactions

FIG. 3, views (a) through (c) show example aspects of a fault-tolerantsequence design, according to some embodiments of the presentdisclosure. View (a) is a diagram representing the interplay betweendisorder, interactions, and control errors in different quantum systems,according to some embodiments. As shown in FIG. 3, view (a), quantumsystems may exhibit some combination of disorder 302, control error 304,and inter-spin interactions 306, with various amounts. For example, adisorder-dominated system is shown as System A 310 and aninteraction-dominated system is shown as System B 320. FIG. 3, view (b)shows example tasks for quantum systems, according to some embodiments.For example, quantum systems can be used for protecting quantuminformation, quantum sensing (e.g., of magnetic fields), and/orsimulation (e.g. as a quantum computer or simulation programmed into aquantum state and evolution). FIG. 3, view (c) shows an exampleoptimized pulse sequence designed to target both the limitations of agiven physical system (FIG. 3, view (a)) and the desired operationmodality (FIG. 3, view (b)), according to some embodiments. As shown inFIG. 3, view (c), the example pulse sequence comprises pulses P₁-P₆ areapplied with time delays τ₁−τ₇ repeated N times.

Example Pulse Imperfections

Without being bound by theory, for purposes of understanding certainexample pulse imperfections, in some embodiments a general form of aHamiltonian can be given by:

$\begin{matrix}{{H(\theta)} = {{U_{0}^{\dagger}{U(\theta)}^{\dagger}H_{f}{U(\theta)}U_{0}} = {{\sum\limits_{i}{\Delta\;{i\left\lbrack {{S_{i}^{z}\cos\;\theta} + {S_{i}^{y}\sin\theta}} \right\rbrack}}} + {\sum\limits_{i,j}{J_{ij}^{{ex},i}\left\lbrack {{\left( {{S_{i}^{x}S_{j}^{y}} - {S_{i}^{y}S_{j}^{x}}} \right)\cos\;\theta} + {\left( {{{- S_{i}^{x}}S_{j}^{z}} + {S_{i}^{z}s_{j}^{x}}} \right)\sin\;\theta}} \right\rbrack}} + {\sum\limits_{i,j}{J_{ij}^{{ex},r}{{\overset{\rightarrow}{S}}_{i} \cdot {\overset{\rightarrow}{S}}_{j}}}} + {\sum\limits_{i,j}{\left( {J_{ij}^{is} - J_{ij}^{{ex},r}} \right)\left\lbrack {{S_{i}^{z}S_{j}^{z}\cos^{2}\theta} + {S_{i}^{y}S_{j}^{y}\sin^{2}\theta}} \right\rbrack}} + {\sum\limits_{i,j}{\frac{1}{2}\left( {J_{ij}^{is} - J_{ij}^{{ex},r}} \right)\left( {{S_{i}^{y}S_{j}^{z}} + {S_{i}^{z}S_{j}^{y}}} \right){{\sin\left( {2\theta} \right)}.}}}}}} & (29)\end{matrix}$

Finite Pulse Width Effects:

Without being bound by theory, according to some embodiments, thederivation of the average Hamiltonian for a π/2 pulse with finite pulsewidth can be given as follows. According to some embodiments, in anexample in which the initial toggling frame with {tilde over(S)}^(x)(0)=S^(x),{tilde over (S)}^(y)(0)=S^(y),{tilde over(S)}^(z)(0)=S^(z) is rotated, such that the rotation brings the framefrom +z into +y, in the interaction picture relative to theinstantaneous control pulse, the operators are transformed as {tildeover (S)}^(x)(θ)=S^(x),{tilde over (S)}^(y)(θ)=S^(y) cos θ−S^(z) sin θ,{tilde over (S)}^(z)θ=S^(z) cos (θ)+S^(y) sin θ. Plugging this into theform of the average Hamiltonian, the instantaneous interaction pictureHamiltonian discussed above can be found.

Without being bound by theory, the transformation properties of theinteraction during the free evolution period can, in some embodiments,also dictate the response during the finite pulse width evolution. Forexample, the disorder and imaginary spin exchange terms, which transformlinearly in F_(μk), involve cos θ or sin θ terms and operators that areproportional to the corresponding Hamiltonian terms in the togglingframes preceding and following the pulse. Meanwhile, the real spinexchange and Ising terms, which transform quadratically in F_(μk),involve similar terms with cos² θ and sin² θ prefactors.

Analysis of Rotation Axis Errors:

In some embodiments, incorporation of axis errors depends on details ofexperimental implementation. In some embodiments, the Hamiltonian isdescribed by the cross product of the rotation angle error and disorder.In some embodiments, this obtains results conditioning.

Example Three-Body Interactions

Without being bound by theory, example decoupling conditions forthree-body interactions are described below, according to someembodiments. Without being bound by theory, in the limit of idealpulses, the decoupling conditions described throughout the presentdisclosure can partially or fully suppress dynamics under any three bodyinteractions for a polarized initial state, according to someembodiments. An example extension of the formalism can account forfinite pulse duration effects in the presence of three-bodyinteractions.

Some physical systems only involve two-body interactions, while othersystems can involve interactions involving more particles, which canlead to a number of physical phenomena, in some embodiments. Forexample, fractional quantum Hall state wavefunctions can appear as theground state of Hamiltonians involving three-body interactions, andother examples of topological phases and spin liquids can be constructedas ground states of such many-spin Hamiltonians. Other examples includecold molecules, superconducting qubits, and a higher-order term in theMagnus expansion of a system with only two-body interactions. Thus,transformation of three-body interactions under periodic driving and theapplication of the disclosed protocol are discussed below.

First, to the complete control and engineering of such interactions,example conditions for dynamical decoupling of such interactions tosuppress any dynamics for a polarized initial state are disclosed,according to some embodiments. For example, interactions under thesecular approximation can be considered such that all terms in theHamiltonian commute with a global magnetic field in the 2-direction.

Ideal Pulse Limit:

without being bound by theory, before discussing example details of thethree-spin coupling, the following nonlimiting lemma for interactionsunder the secular approximation can be proved.

Lemma: For any interaction under the secular approximation, averagingunder the spin-½ single qubit Clifford group can be equivalent toaveraging under unitary operators that cover the six axis directions.

Proof: without being bound by theory, the lemma can be proved by firstconsidering a generic n-body interaction Hamiltonian H and a set ofunitary operators U_(k) (k=1, 2, . . . ,N) to average over, then

$\begin{matrix}{{H_{ave} = {\frac{1}{N}{\sum\limits_{k}{\left( U_{k}^{\dagger} \right)^{\otimes n}HU_{k}^{\otimes n}}}}}.} & (30)\end{matrix}$The elements of the Clifford group can be grouped by how they transformthe S^(z) operator, such that each set contains elements that satisfyU^(†)S^(z)U=(−1)^(v)S^(μ), but U^(†)S^(x)U can take four distinct valuesthat are orthogonal to the S^(μ) operator direction. In the disclosedframe representation, these can correspond to a single term specified by(−1)^(v)S^(μ). Thus, proving the lemma reduces to proving that the fourClifford elements above give identical Hamiltonians.

Given two elements U₁ and U₂ in the set, there can exist a rotationU_(z) around the {circumflex over (z)} axis such that U₁=U_(z)U₂ (thisleaves the interaction picture {tilde over (S)}^(z) invariant, butchanges {tilde over (S)}^(x)). The Hamiltonian under conjugation by U₁can then be given byH ₁=(U ₁ ^(†))^(⊗n) HU ₁ ^(⊗n)=(U ₂ ^(†))^(⊗n)(U _(z) ^(†))^(⊗n) HU _(z)^(⊗n) U ₂ ^(⊗n).  (31)Under the secular approximation, a rotation around the {circumflex over(z)} axis does not modify the Hamiltonian, since the Hamiltonian cancommute with the global S^(z). Consequently, the preceding averageHamiltonian after conjugating by U₁ can be equal to that conjugating byU₂, proving the lemma.

In some embodiments, without being bound by theory, given this lemma,mathematical results from unitary t-designs can show that aftersymmetrizing along the six axis directions, as can be guaranteed by theconditions described in the present disclosure, a polarized initialstate can be an eigenstate of the resulting symmetrized Hamiltonian.

In some embodiments, a Unitary t-design is a set of unitary operators{U_(k)}, such that for every polynomial P_((t,t))(U) of degree at most tin U and at most t in U*, the average over {U_(k)} can be equivalent tothe average over the Haar measure of all unitaries of the samedimension. Without being bound by theory, this can imply that for anyN-body operator

with N≤t, the expectation values after performing a global conjugationby the unitaries are the same

$\begin{matrix}{{{\frac{1}{K}{\sum\limits_{k = 1}^{K}{\left( U_{k}^{\dagger} \right)^{\otimes N}\mathcal{O}\; U_{k}^{\otimes N}}}} = {{\int_{u{(2)}}{d{U\left( U^{\dagger} \right)}^{\otimes N}\mathcal{O}\; U^{\otimes N}}}\overset{\Delta}{=}\mathcal{O}_{U}}},} & (32)\end{matrix}$where

(2) is the unitary group of dimension 2. This can imply that the effectof averaging over the finite set of unitary operators {U_(k)} can beequivalent to averaging over all random unitaries, up to observables oforder t.

Without being bound by theory, the symmetrizing properties of theright-hand-side of Equation (32), where the average is taken over allelements of the unitary group over the Haar measure, can imply that theresulting average operator

_(U) can only contain terms proportional to elements of the symmetricgroup S_(t) of order t, according to some embodiments. This can bebecause all other terms can be transformed and symmetrized out by theaverage, but elements of the symmetric group, which permute the labelsof the states, can be invariant due to the product structure of theunitary operators, in some embodiments.

The Clifford group can form a unitary 3-design. Without being bound bytheory, in some embodiments, this, combined with the lemma presentedabove, this can imply that for any sequence that equally covers all sixaxis directions, all interactions involving three particles or less canbe symmetrized into a form that only contains terms proportional toelements of the symmetric group. In some embodiments, any initial statewith all spins polarized in the same direction can then be an eigenstateof this symmetrized interaction, since this state can be invariant underany permutation of the elements. Correspondingly, a polarized initialstate does not experience decoherence under this interaction, accordingto some embodiments.

As a nonlimiting example of this result, the interactionH_(int)=J(S^(x)S^(y)S^(z))−S^(y)S^(x)S^(z) can be considered. Thesymmetrized Hamiltonian can be calculated to be

${H_{int} = {{\frac{J}{3}\sum_{i,j,k}} \in_{ijk}{S^{i}S^{j}S^{k}}}},$where ∈_(ijk) is the Levi-Civita symbol. Without being bound by theory,in some embodiments it has been verified that any globally polarizedinitial state can be an eigenstate of the symmetrized Hamiltonian witheigenvalue 0.

Without being bound by theory, in some embodiments, the precedingprocedure can also suggest that four-body interactions can still inducedynamics after symmetrization, since the Clifford group is not a unitary4-design. It has been verified that this is the case by considering thesymmetrized interaction (S^(x))^(⊗4)+(S^(y))^(⊗4)_(S^(z))^(⊗4), which isfound to act nontrivially on a generic polarized initial state.

Finite Pulse Duration Effects:

Without being bound by theory, in some embodiments, to achieve robustdynamical decoupling of the interaction Hamiltonian, finite pulseduration effects for three-body interactions can be considered. For thispurpose, the disclosed representation of the transformation propertiesof an interaction term based on the matrix F can be used, which canprovide a complete description of the Hamiltonian under the secularapproximation, according to some embodiments. Without being bound bytheory, in some embodiments, considering up to three-body interactions,similar to equations (11-14), the transformation properties of a genericn-body interaction can be written as a polynomial in F_(μk):

$\begin{matrix}{{\overset{\sim}{H} = {\sum\limits_{l = 0}^{3}{F_{\mu k}^{l}G_{l}}}},} & (34)\end{matrix}$where G_(l) includes the spin operators and remaining coefficients. Forexample, without being bound by theory, any nontrivial three-bodyinteraction can be written as a sum of terms, each composed of a productof nontrivial Pauli operators. The S^(x) and S^(y) operators can changethe total magnetization by 1, so that under the secular approximationthere can be an even number of them. Consequently, the interaction caneither be of the form S^(z)S^(z)S^(z), or involve the tensor product ofan S^(z) operator and a polarization-conserving two-body operator, whichcan be S^(x)S^(x)+S^(y)S^(y) or i(S^(x)S^(y)−S^(y)S^(x)). Without beingbound by theory, in some embodiments, each term can be written as aproduct of individual pieces that transform as F_(μk). For example, thethree-body interaction

$\begin{matrix}{{{i\left( {{S^{X}S^{y}} - {S^{y}S^{x}}} \right)}S^{z}} = {i{\sum\limits_{\mu\upsilon\sigma}{\left\lbrack {\epsilon_{\mu\;{\upsilon\sigma}}{F_{\mu k}\left( {{S_{i}^{\upsilon}S_{j}^{\sigma}} - {S_{i}^{\sigma}S_{j}^{\upsilon}}} \right)}} \right\rbrack\left\lbrack {F_{\mu k}S^{\mu}} \right\rbrack}}}} & (35)\end{matrix}$In some three-body interaction embodiments, the first step is to controldecoupling.Comparison of Sequence Performance

FIGS. 7A-7C show example dynamics of various decoupling sequences,according to some embodiments. The sensitivity to external fields of aspin-based sensor can scales with the coherence time, T₂, asη_(v)∝1/√{square root over (T₂)}. Consequently, extending T₂ is one wayto achieve the best performance in a device. The coherence time T₂ ofthe spin ensemble can be obtained by monitoring the dephasing of thetotal spin magnetization from a polarized initial state. FIG. 7D shows adifferential measurement that can be used to probe the spin coherence,with an additional π pulse on the second repetition to rejectcommon-mode noise. In particular, NVs are initialized using pulsed laserexcitation at 532 nm (top trace) and read out through emitted photonsdetected by a single photon counting module (second trace from the top).N repetitions of a sensing sequence unit of length T (bottom trace) andrepeat the same measurement with an additional π pulse (second tracefrom the bottom) acting on the NV centers for differential readout ofthe spin state.

FIG. 7A illustrates the details of different pulse sequences, XY-8,Sequation A and Sequation B, composed of π/2 and π rotations along 2 and9 axes, according to some embodiments. In FIG. 7A, white and black barsin Sequation B indicate rotation pulses along the X and Y axis,respectively. The sequence can be described by pulses P_(i) and thetime-dependent frame transformation of the system between the pulses, asshown in FIG. 7B. The orientation of each rotated frame is highlightedby a bolded axis that points along the {circumflex over (z)} axis of thefixed external reference frame. Decoupling sequences can be designed byimposing average Hamiltonian conditions on the evolution of the boldedaxis. For example, the effects of disorder can be cancelled byimplementing an echo-like evolution, +{circumflex over (μ)}→−{circumflexover (μ)} where μ=x, y, z (top row), and interactions are symmetrized byequal evolution in each of the {circumflex over (x)}, ŷ, {circumflexover (z)} and {circumflex over (z)} axes in the transformed frames(bottom row). Additionally, the pulse sequence can be designed tomutually correct rotation angle errors and finite-pulse effects.

FIG. 7C shows the experimental performance of different sequences,according to some non-limiting experimental observations. In particular,FIG. 7C shows the coherence of an ensemble as a function of time. Inaddition, ensemble spin coherence is probed under different sequencesdesigned to decouple various types of imperfections (table inset in FIG.7C). The example decoherence profiles can be fit with a stretchedexponential e^(−(t/T) ² ⁾ ^(α) (solid curves) to extract the coherencetime T₂ for each sequence. Simple spin-echo (crosses), XY-8 (circles)and Sequation A (diamonds) show T₂=0.98(2) μs, 1.6(1) μs, and 2.8(1) μswith α=1.5(1), 0.66(2), and 0.61(3), respectively. Sequation B(squares), designed to correct for leading order effects ofinteractions, disorder and control imperfections, performs best atT₂=7.9(2) μs with α=0.75(2). The coherence time of Sequation B isindependent of the initial state prepared along {circumflex over (x)}, ŷand {circumflex over (z)} axes, as shown in FIG. 7B.

As shown in FIG. 7C, in some examples of observed dense NV ensemble, thespin echo T₂ can be limited to only 1.0 μs, as shown, for example, inFIG. 7C, plotted with X's. One method to extend T₂ beyond the spin echois the XY-8 dynamical decoupling sequence, consisting of equally spacedπ pulses along the 2 and 9 axes, as shown in the XY-8 pulse sequence inFIG. 7A. However, the XY-8 sequence only provides a small improvement,leading to T₂=1.6 μs (FIG. 7C, squares), despite its capability todecouple sensor spins from environment-induced disorder. Without beingbound by theory, this can be explained by the fact that the π rotationsin the XY-8 sequence, which globally flip all spins simultaneously,leave interactions between spins unchanged. Therefore, strong spin-spininteractions in a spin system limit the observed XY-8 coherence time.

Example Hamiltonian of a Spin Ensemble

Without being bound by theory, in some embodiments, a spin ensemble,including the applied control fields and the external sensing field, candescribed by the HamiltonianH=H ₀ +H _(Ω)(t)+H _(AC)(t),  (36)where the internal system Hamiltonian is

${H_{0} = {{\Sigma_{i}h_{i}S_{i}^{Z}} + {\Sigma_{ij}\frac{J_{ij}}{r_{ij}^{3}}\left( {{S_{i}^{x}S_{j}^{x}} + {S_{i}^{y}S_{j}^{y}} - {S_{i}^{Z}S_{j}^{Z}}} \right)}}},$containing both on-site disorder and long-range dipolar interactionsbetween spins. Global time-dependent control pulses used for spinmanipulations can be given by H_(Ω)(t)=Σ_(i)(Ω_(i) ^(x)(t)S_(i)^(x)+Ω_(i) ^(y)(t)S_(i) ^(y)) and an external target signal byH_(AC)(t)=γ_(NV)B_(AC) cos(2πf_(AC)t−ϕ)Σ_(i)S_(i) ^(z). In this exampleembodiment, S_(i) ^(μ)(μ=x, y, z) are spin-½ operators, h_(i) is arandom on-site disorder potential following a normal distribution withstandard deviation W=(2π) 4.0 MHz, J_(ij)/r_(ij) ³ is the anisotropicdipolar interaction strength between two spins of the samecrystallographic orientation at a distance r_(ij) with an averagestrength of J=(2π) 0.1 MHz at a typical separation. The global controlamplitudes, Ω_(i) ^(x,y)(t), can be position-dependent due to spatialfield inhomogeneities, γ_(NV) is the gyromagnetic ratio of the NVcenter, and B_(AC), f_(AC) and ϕ are the amplitude, frequency and phaseof the target AC signal, respectively.Example Design Protocol for Designing Pulse Sequences

Without being bound by theory, in order to achieve a significantextension of the spin coherence time T₂ in the presence of interactions,a protocol can be used to design pulse sequences that can suppressinteractions and disorder, and can be additionally fault-tolerantagainst rotation angle errors and finite pulse duration effects,according to some embodiments. For example, an average Hamiltoniantheory can be applied to engineer the system Hamiltonian through pulsedperiodic manipulation of the spins. In some embodiments, a sequencecomposed of n equidistant control pulses {P_(k); k=1, 2, . . . , n} withspacing τ can define a unitary operator over a period T given by U(T)=

. . .

. Here, {tilde over (H)}_(k)=U_(k) ^(†)H₀U_(k) are the transformedHamiltonians in the interaction picture with U_(k)=P_(k) . . . P₂P₁ andU_(n)=1. In some embodiments, if the pulse spacing τ is much shorterthan the corresponding timescales of the system

$\left( {{\tau ⪡ \frac{1}{W}},\frac{1}{J},} \right.$or the dynamics of the many-body state |ψ(t)

can be governed by an effective average Hamiltonian

$\begin{matrix}{{H_{ave} = {\frac{1}{T}{\sum\limits_{k = 1}^{n}{\overset{\sim}{H}}_{k}}}},} & (37)\end{matrix}$such that at |ψ(t)

=

(T)^(N)|ψ(0)

≈e^(iH) ^(ave) ^(t)|ψ(0)

times t=NT with integer N.

In some embodiments, a pulse sequence can be developed that generates adesirable form of H_(ave) from the original H₀ intrinsic to the system.In some embodiments, an ideal pulse sequence can be designed to yieldH_(ave)=0 and thus preserve quantum coherence for all initial states. Insome embodiments where interactions cannot be fully cancelled in the NVensemble if the system is subject to global spin rotations only, H₀ canbe engineered to transform into an average Hamiltonian of the Heisenberginteraction form,

$H_{ave} \propto {\Sigma_{ij}\frac{J_{ij}}{r_{ij}^{3}}{\left( {{\overset{\rightarrow}{S}}_{i} \cdot {\overset{\rightarrow}{S}}_{j}} \right).}}$Polarized states are eigenstates of this H_(ave), and we thereforeexpect the coherence of our polarized initial states to be protected.

In some embodiments, without being bound by theory, Hamiltonianengineering can be understood as the result of frame transformations,which rotate the interaction-picture operators (FIG. 7B): for example, aπ pulse flips S_(i) ^(z)→−S_(i) ^(z), while a π/2 pulse rotates S_(i)^(z)→±S_(i) ^(x,y), with the final sign and axis direction determined bythe rotation axis and sign of the pulse. In some embodiments, theaverage Hamiltonian is uniquely specified by the transformations of theS^(z) operator in the interaction picture. Without being bound bytheory, in some embodiments, using this technique to suppress theeffects of disorder, a sequence can be designed such that the operatorhas an equal duration of evolution in the positive and negativedirection for each axis, effectively producing a spin echo along allthree axes. To transform the anisotropic dipolar interaction into anisotropic Heisenberg form, the {circumflex over (x)}, ŷ, {circumflexover (z)} axis directions can be required to appear for an equal amountof time, leading to symmetrization. These two example conditions canallow the choice of decoupling prioritization that best suits the systemproperties: for spin ensemble where disorder is dominant (W>>J), theecho operation can be performed at a higher rate than interactionsymmetrization.

Without being bound by theory, in some embodiments the Hamiltoniananalysis presented in the three preceding paragraphs has assumedinfinitely short pulses with no errors. In some example applications,such an ideal setting is unrealistic, as imperfections of the averageHamiltonian, δH_(ave), can arise from control errors and the finitepulse width. However, the same description of the average Hamiltonian interms of the interaction-picture S^(z) operator also allows simplealgebraic conditions to be written for suppressing the dominant effectsof disorder and interactions during the finite pulses as well as theimpact of rotation angle errors. Thus, pulse sequences can besystematically generated that are fault-tolerant to all of theseimperfections, to yield a pure Heisenberg Hamiltonian with δH_(ave)=0.

Without being bound by theory, to illustrate the importance ofembodiments of the disclosed robust sequence design, the decouplingefficiency of a pulse sequence (Sequation A in FIG. 7A) can be observedthat decouples disorder and interactions in the ideal pulse limit, butdoes not suppress all pulse-related imperfections. This is ageneralization of the WAHUHA sequence, where in addition to the existinginteraction symmetrization, the effects of disorder can also bedecoupled for finite pulse durations. In some applications, comparedwith XY-8, only a modest increase in coherence time is observed, T₂=2.8μs (FIG. 7C, diamonds), indicating that the remaining imperfections playan important role. To address these imperfections, the formalismdescribed throughout the present disclosure can be used to designSequation B, as shown in FIG. 7A, which realizes pure Heisenberginteractions at the average Hamiltonian level. Without being bound bytheory, in some embodiments, due to fault-tolerance of Sequation Bagainst all leading-order effects, it shows a significant extension ofcoherence time compared to the sequences described above, reachingT₂=7.9 μs (FIG. 7C, squares). Moreover, the observed coherence time isindependent of the initial state. Without being bound by theory, in someembodiments, while the leading-order errors are fully suppressed asdesigned, numerical simulations indicate that the T₂ reached here islimited by higher-order terms originating from the non-commuting natureof transformed Hamiltonians {tilde over (H)}_(k) at different times.

Modulation Functions and Optimal Vector Sensing

Without being bound by theory, in some embodiments, the sensitivity of asequence to an external signal field can be influenced by both by theachievable coherence time and the spectral response of the sequence. ForAC magnetic field sensing, one approach to improving sensitivity is toimplement a periodic inversion of the spin operator between S^(z) and−S^(z) in the interaction picture, driven by a train of equidistant πpulses at a separation of

$\frac{1}{2f_{AC}}.$This modulation can allow for the cumulative precession of the sensorspin when the AC signal sign change coincides with the frame inversion,resulting in high sensitivity to a signal field at f_(AC).

Without being bound by theory, example decoupling sequences developedusing the disclosed protocol explore all three frame directions S^(x),S^(y), S^(z). Similar techniques can be used to design a sensorresonance at the target frequency. In some example embodiments, thecriterion for AC selectivity includes periodic frame inversions in eachof the three axes. This can be satisfied while at the same timepreserving the desired H_(ave), which produces long coherence timesthrough robustness to disorder, interactions and pulse imperfections.

FIG. 8A is an illustration showing how to achieve periodic frameinversions while maintaining desired H_(ave) for the example Sequation Bdescribed above. In particular, FIG. 8A shows a Pulse sequence (top row)and three-axis time-domain modulation functions (bottom curves) for thefirst 14 free evolution times of Sequation B. White and black bars inSequation B indicate rotation pulses along the X and Y axis,respectively. In some embodiments, the modulation period along each axisis synchronized to an AC sensing signal (center sinusoidal curve). Insome embodiments, the pulses lead to periodic changes in the sign andorientation of the interaction-picture S^(z) operator, depicted by thetime-domain modulation functions for each axis direction, F_(x), F_(y)and F_(z). As shown in the example shown in FIG. 8A, all three functionsexhibit a phase-locked periodic sign modulation, indicating thatSequation B can be resonant with an AC signal oscillating with the samemodulation period. Performing the Fourier transforms of F_(μ)(μ=x, y,z), obtains the detailed resonance characteristics of the pulsesequence:

$\begin{matrix}{{{{\overset{˜}{F}}_{\mu}(f)} = {{{{{\overset{˜}{F}}_{\mu}(f)}}e^{i{\overset{\sim}{\;\phi}}_{\mu}{(f)}}} = {\frac{1}{NT}{\int_{0}^{NT}{e^{{- i}2\pi\; f\; t^{\prime}}{F_{\mu}\left( t^{\prime} \right)}{dt}^{\prime}}}}}},} & (38)\end{matrix}$where N is the sequence repetition number, T is the duration of theFloquet period, and {tilde over (Ø)}_(μ)(f) is the spectral phasecapturing the relative phase difference between different axes.

FIG. 8B shows the calculated spectral intensities along different axes,|{tilde over (F)}_(x)(f)|², |{tilde over (F)}_(y)(f)|² and |{tilde over(F)}_(z)(f)|² as well as the total intensity |{tilde over (F)}_(t)(f)|²,according to some embodiments. In particular, FIG. 8B shows thefrequency-domain modulation function |{tilde over (F)}_(x,y,z)(f)|² forSequation B, with pulse spacing τ=25 ns and π/2-pulse width τ_(π/2)=10ns. The total strength |{tilde over (F)}_(t)(f)|² is obtained, in theexample of FIG. 8B from individual axis amplitudes {tilde over(F)}_(x,y,z)(f) in consideration of their relative phases in thefrequency domain. The principal resonance is highlighted by an arrow,yielding maximum sensitivity. At the dominant resonance of the totalintensity (arrow in FIG. 8B), all three axes are phase-locked to eachother, leading to constructive phase acquisition and high sensitivity.

In addition to designing the frequency response of the sequence, in someembodiments optimal spin initialization and readout procedures thatprovide the best sensitivity can be developed. Without being bound bytheory, generalizing the average Hamiltonian analysis to incorporate theAC signal field, the average sensing Hamiltonian can be given by:

$\begin{matrix}{{H_{{ave},{A\; C}} = {{\gamma_{NV}B_{AC}{\sum\limits_{i}{{Re}\left\lbrack {\sum\limits_{{\mu = x},y,z}{{{\overset{˜}{F}}_{u}\left( f_{AC} \right)}S_{i}^{\mu}e^{i\;\phi}}} \right\rbrack}}} = {\gamma_{NV}{\sum\limits_{i}{{\overset{\rightarrow}{B}}_{eff} \cdot {\overset{\rightarrow}{S}}_{i}}}}}},} & (39)\end{matrix}$where {right arrow over (B)}_(eff) is an effective magnetic field vectorin the interaction picture which appears static to the driven spins. Insuch an example, the spins undergo a precession around {right arrow over(B)}_(eff), with the field orientation determined by thefrequency-domain modulation functions {tilde over (F)}_(μ) and thestrength by the total intensity |{tilde over (F)}_(t)| value. For theXY-8 sequence, {right arrow over (B)}_(eff) ∝([0, 0, 1] with |{tildeover (F)}_(x)|=|{tilde over (F)}_(y)|=0 and |{tilde over (F)}_(z)|≠0. Inthis case, spin initialization into the {circumflex over (x)} or ŷ axis(perpendicular to {right arrow over (B)}_(eff)) provides bestsensitivity. In some embodiments, implementing the disclosedinteraction-symmetrization means that only ⅓ of the total sensing timecan be spent along any given axis, resulting in {right arrow over(B)}_(eff) ∝[0, 0, ⅓] and a significant loss of sensitivity for anyinteraction decoupling sequence. This can be overcome by making use ofthe vector nature of {right arrow over (B)}_(eff) to achieve optimalsensitivity. Without being bound by theory, in the example Sequation B,the signal at the principal resonance f_(AC) gives rise to |{tilde over(F)}_(x)|=|{tilde over (F)}_(y)|=|{tilde over (F)}_(z)| with {tilde over(Ø)}_(x)={tilde over (Ø)}_(y)={tilde over (Ø)}_(z), leading to {rightarrow over (B)}_(eff)∝[⅓,⅓,⅓]. The resulting strength |{right arrow over(B)}_(eff)| of Sequation B can be limited to 1/√{square root over (3)}of the value reached in XY-8, due to interaction symmetrization. Underthis constraint, sensitivity can be maximized by initializing the spinsin a plane substantially perpendicular to the [⅓,⅓,⅓]-direction to allowthe largest precession orbit, thereby optimally accumulating phase fromthe signal. This perpendicular direction is shown in FIG. 8C. Inparticular, FIG. 8C shows an illustration of the effective magneticfield created by Seq B at the principal resonance, according to someembodiments. In the average Hamiltonian picture, the {circumflex over(z)}-direction sensing field in the external reference frame transformsto the [1,1,1]-direction field {right arrow over (B)}_(eff) in theeffective spin frame with a reduced strength (18). Optimal sensitivitycan be achieved, in some embodiments, by initializing the spins into theplane perpendicular to the effective magnetic field direction. Thisoptimal state preparation can allow the spins to precess along thetrajectory with the largest contrast (dashed line 102). For comparison,the precession evolution for initialization to the conventional{circumflex over (x)} axis is shown as dashed line 104.

In some embodiments, for spin-state readout, a rotation axis [−1, 1, 0]and an angle of cos⁻¹(√{square root over (⅔)}) to rotate the precessionplane parallel to the {circumflex over (z)} axis, which can providemaximal signal contrast. FIG. 8D shows sensing resonance spectra nearthe principal resonance. The optimal initialization (circles) showsgreater contrast than the {circumflex over (x)}-axis initialization(squares). Square and circular markers indicate experimental data andsolid lines denote theoretical predictions calculated from thefrequency-domain modulation functions. As shown in FIG. 8D, a largercontrast can be observed when spins are initialized in the optimaldirection along [1, 1, −2] (orthogonal to [1, 1, 1]), compared to aninitialization in the {right arrow over (x)} direction.

EXAMPLE IMPLEMENTATIONS

As discussed above, in some embodiments, the disclosed Sequation Bdeveloped in accordance with the disclosed protocol can demonstrate longcoherence times and optimal initialization and measurement conditions.Without being found by theory, this section characterizes thesensitivity of example implementations of Sequation B, and compares itto the XY-8 sequence.

Without being bound by theory, magnetic field sensitivity can bedescribed as the minimum detectable signal amplitude per unit time, andcan be given by

$\eta = {\frac{\sigma_{s}}{\left| {d{S/d}B_{AC}} \right|}.}$Here, S is the sensor signal contrast, σ_(s) is the uncertainty of S forone second of averaging, and |dS/dB_(AC)| is the gradient of S withrespect to the field amplitude B_(AC). For sensing non-limiting examplemeasurements, the following parameter values can be selected: a greenlaser power of 75 μW, repolarization duration of 6 μs, andphoton-counting period of 1.2 μs to optimize the absolute sensitivity.For a comparison of the nonlimiting example sensing performances of thetwo sequences (Sequation B and XY-8), the interrogation time can beindependently optimized, the number of periods the sequence can berepeated, as well as the phase and frequency of the AC signal.

FIG. 9A shows the measured contrast S as a function of B_(AC) underSequation B and XY-8, according to some non-limiting exampleimplementations of embodiments of the present disclosure. In particular,FIG. 9A shows example observed spin contrast as a function of ACmagnetic field strength, for the XY-8 sequence (circles) and for animplementation of Sequation B (squares). The example fit is a sinusoidaloscillation with an exponentially decaying profile. As shown in FIG. 9A,this implementation of Seq B shows a steeper slope at zero field,indicating that it is more sensitive than XY-8 to the example sensedexternal field. The example difference observed in contrast modulationamplitude between the sequences can be associated with differentinterrogation times t that are independently optimized to achievemaximal sensitivity (with t=2.16 μs for XY-8 and t=6.52 μs for SequationB). The resonant AC signal can induce a precession of the sensors pins,resulting in sinusoidal modulations of S as a function of B_(AC). Forthe example initial states and readout directions, the maximum slope|dS/dB_(AC)| is obtained, in this example, at zero signal field usingsine magnetometry. This implementation of Sequation B shows asignificantly steeper slope than XY-8, indicating that it is moresensitive to the external signal. The example higher contrast and fasteroscillations of this implementation of Sequation B result from acombination of its significantly improved coherence time and optimalstate preparation and readout schemes, despite a relative sensitivitypenalty associated with the reduced strength of |{right arrow over(B)}_(eff)|.

FIG. 9B shows the sensitivity scaling with interrogation time, accordingto some embodiments. In particular, FIG. 9B shows extracted absolutesensitivity η and volume-normalized sensitivity η_(V)=η√{square rootover (V)} with sensing volume V=0.018 μm³ as a function of interrogationtime t, according to some example implementations. In FIG. 9B, squaresindicate implementations of implementation of Sequation B, while circlesindicate implementations of XYY-8. A comparison of the two sequences attheir respective optimal sensing times reveals that Sequation Boutperforms XY-8 by ˜30%. Without being bound by theory, the exampleplotted in FIG. 9B has good agreement with the theoretical prediction

${{\eta(t)} \propto {e^{{({t/T_{2}})}^{\alpha}}\frac{\sqrt{t + T_{p}}}{t}}},$where α is the exponent of the decoherence profile and T_(p) is thepreparation time needed for sensor initialization and readout.

In some embodiments, by identifying the minimum η value at an optimalsensing time, the example best volume-normalized sensitivity,η_(v)=η√{square root over (V)}, can be extracted for each sequence.Example implementations of Sequation B, designed with the fault-tolerantoptimal sensing approach described throughout the present disclosure,reaches more than 30% improvement in sensitivity over the XY-8 sequence.With these enhancements, example implementations demonstrate η_(V)=28(1)nT·μ.m^(3/2)√{square root over (Hz)} for Sequation B.

FIGS. 10A-10E show example implementations of fault-tolerant sequencesusing dense NV centers in diamond, according to some embodiments. Thisexample implementation uses a DC external magnetic field to split thespin-1 ground state and address the |0

and |−1

levels (see FIG. 1B). The NV electronic spin state can be opticallyinitialized and readout, and manipulated using microwave driving, forexample, using the apparatus shown in FIG. 1A. A nanobeam sample is usedto limit the probing volume and improve the homogeneity of controlfields. The tested sample contains a high density (45 ppm) of NVcenters, such that the magnetic dipolar interaction strength issignificantly larger than extrinsic decoherence rates, and the coherencetimes for an XY-8 sequence are interaction limited. Without being boundby theory, in some embodiments this sample confirms various aspects ofthe disclosed design protocol. The sample has relatively large on-sitedisorder (˜4 MHz) with modest interaction strengths (˜100 kHz within asingle NV lattice orientation), corresponding to the regime in FIG. 10Bwhere Sequation B is expected to perform significantly better thanSequation A (Cory-48). This is confirmed in the example of FIG. 10A,where Sequation B shows a significantly longer coherence time comparedto Sequation A.

FIG. 10A is a graph showing a comparison of spin coherence decay underthe application of Sequation A (Cory-48, squares) and Sequation B(circles), according to some embodiments. Sequation B, tailored fordisorder-dominated systems, outperforms Sequation A in the black diamondsystem.

FIG. 10B shows graphs showing sequence robustness conditions withSequation B (left view) and a slightly modified counterpart Sequation D(right view), which only differs in patching frames and no longercancels rotation angle errors, according to some embodiments. FIG. 10Cis a graph showing modulation frequency for the two sequences B and D asa result of spin rotation errors, as a function of relative rotationangle error, according to some embodiments. FIGS. 10B and 10C can beused to compare the robustness of two different sequences B and Dagainst systematic rotation angle deviations to illustrate thedecoupling criteria in the disclosed protocol. Here, Sequation Bsuppresses rotation angle errors, while Sequation D is almost identicalto Sequation B, except the patching frames are permuted in a way toresult in a residual rotation angle error. FIG. 10B illustrates thedecay profile of the sequence when the rotation angle is chosen to be92.5% of the correct rotation angle (white circle in FIG. 10C);Sequation B does not show any oscillations, while Sequation D showspronounced oscillations over time. This behavior is further confirmed inFIG. 10C, the effective rotation frequency of the spin coherence as afunction of the systematic rotation angle deviation is extracted. WhileSequation B does not show any oscillations, Sequation D shows a lineardependence of modulation frequency as the rotation angle deviation isincreased, indicating that it is not robust against such perturbations.

Sequation D shows pronounced oscillations, while Sequation B is muchmore robust. FIG. 10D shows graphs showing the frequency response ofsensing Sequation C for initial states {right arrow over (x)}: (1, 0, 0)and optimal: (−1, −1, 2)/√{square root over (6)} in squares andtriangles, respectively, according to some embodiments. In particular,FIG. 10D shows the contrast change under the presence of an AC magneticfield with varying frequency, for Sequation C discussed throughout thepresent disclosure. As confirmed by FIG. 10D, the resonances {circlearound (1)} and {circle around (2)} have comparable field strengthsalong each axis, but due to the fact that resonance {circle around (1)}has aligned phases while resonance {circle around (2)} does not, thetotal field strength is larger for the former. For example, as shown inFIG. 10D, the top view shows the contrast change due to the sensingsignal for an {right arrow over (x)} initial state, where the tworesonances have comparable amplitude. However, as can be seen in thebottom view, the sensing signal induces a much larger change in contrastfor the [1,1,1] initialization direction for resonance {circle around(1)}.

The phase dependence of different axes is further highlighted in FIG.10E, where the contrast is examined as a function of the phase of thesensing signal. FIG. 10E shows graphs showing the phase response of thesensing signal for Sequation C resonance {circle around (1)} (top) andresonance {circle around (2)} (bottom), with initial states x, y, z andoptimal shown in circles, squares, diamonds, and triangles,respectively, according to some embodiments. A120 degree phase shiftbetween individual initial states can be seen for resonance {circlearound (2)} and in-phase response at resonance {circle around (1)},where the optimal initial state provides highest contrast. For resonance{circle around (1)}, the phase is aligned between the resonances, whilefor {circle around (2)}, the curves are out-of-phase from each other interms of their oscillation behavior. Thus, without being bound bytheory, to obtain optimal sensitivity, it is desirable to align not justthe resonance frequency, but also the relative phase of thecontributions from each axis.

Limitations on Sensitivity

One measure of a quantum sensor is the sensor's sensitivity, which canrefer to how well the quantum sensor responds to (i.e., picks up) faintor small signals. In order to improve the sensitivity of a quantumsensor, a dense ensemble of individual sensors can be used to takeadvantage of parallel averaging. For example, without being bound bytheory, the sensitivity enhancement with increasing sensor density canbe described in some example implementations by the standard quantumlimit (SQL) which can take the form ηv∝1/√{square root over (pT₂)},where ηv is the volume-normalized field sensitivity, p is the sensordensity and T₂ is the relevant coherence time. However, the SQL does notalways hold across all densities. For example, in some applications athigh densities, the interactions between sensors can increase, which canlead to a decrease in the spin-spin relaxation time T₂, which is ameasure of the coherence time of a quantum spin. This limit on T₂ canarise from interactions that drive thermalization, a process that canlead to a system losing its coherence over time. Consequently, thesensitivity of conventional quantum sensor applications, which oftenneglect interactions between sensors, can deviate from the SQL beyond acertain critical density and become limited by spin-spin interactions.

Example Application: Efficient Dynamical Decoupling Design

The performance of example applications of pulse sequences designedusing the disclosed protocol are described below. A first exampleapplication applies the disclosed protocol to design different dynamicaldecoupling pulse sequences that are targeted towards the suppression ofdifferent types of dominant effects. Without being bound by theory, thetimescales in which disorder and interaction are decoupled (T_(w),T_(J)) are compared for different sequences. In some embodiments, thesetimescales have an effect on the decoupling properties.

FIG. 6, views A-C show aspects of designing pulsed decoupling sequenceswith robustness to different types of dominant energy scales. FIG. 6,view A is a diagram showing a pulse representation of Sequation A(Cory-48), where axis permutation is performed on the fastest timescale,prioritizing the cancellation of interaction effects. FIG. 6, view B isa diagram showing a pulse representation of Sequation B, where spinechoes are performed on the fastest timescale, prioritizing disorderdecoupling. Frame representations for the first 12 free precessionintervals of each sequence are shown between view A and view B,including the time-scales for interaction decoupling (T_(J)) anddisorder decoupling (T_(W)). FIG. 6, view C is a graph showing T₂-decaytime for the sequences in view A (Sequation A) and view B (Sequation B)as a function of disorder and interaction strength, where a clearcrossover of performance is observed. Pulses are assumed to beinfinitesimally short and τ=20 ns.

In the example of FIG. 6, the decoupling performance of the Cory-48pulse sequence (denoted Sequation A in FIG. 6 view A) is compared tosequence (denoted Sequation B in FIG. 6 view B) design based on thedisclosed protocol, over a range of interaction and disorder strengths.The Cory-48 pulse sequence utilizes the solid-echo pulse block, and maybe viewed as an example in which interactions are cancelled on a rapidtimescale, and disorder is suppressed on a slower timescale (T_(W) ismuch longer than T_(J) in FIG. 6, view A). Additional symmetrization canalso be performed to suppress higher-order effects. The exampleSequation B shown in FIG. 6 incorporates π pulses to echo out disorderon a rapid timescale (T_(W) is short in FIG. 6, view B), while alsousing composite π/2 pulses to suppress all average Hamiltonian effectsin the presence of pulse imperfections. It is designed such that theeffects of disorder due to the finite pulse duration are also echoed outon the fastest timescale, further improving the robustness againston-site disorder.

Without being bound by theory, given these design considerations,example implementations of the Cory-48 pulse sequence perform better inthe regime of large interaction strengths (e.g. NMR), while exampleimplementations of Sequation B to performs better for disorder-dominatedsystems (e.g. electronic spin ensembles).

This is verified in an example application by performing exactdiagonalization simulations of a disordered, interacting ensemble of 12spins, under a variety of disorder and interaction strengths, with eachparameter set averaged over 100 disorder realizations, according to someembodiments. The pulse duration is chosen to be infinitesimally shortand τ=20 ns, so that Jτ and Wτ are the only dimensionless parameters ofthe system. The results are shown in FIG. 6, view C, where a crossoverin performance as measured by T₂ can be observed as the disorder andinteraction strengths are tuned in the system. For the range ofinteraction strengths chosen, for small disorder values (<1 MHz),Cory-48 (Sequation A) has a longer coherence time. However, as thedisorder strength increases, a crossover occurs in which Sequation Bstarts to perform better. Overall, Sequation B has a fairly stableperformance within the range of parameters shown in FIG. 6, view C,while Cory-48 shows a strong susceptibility to disorder. This exampleillustrates how embodiments of the disclosed protocol enable thesystematic design of pulse sequences that are suited for differentsystems, where the pulse sequence design can be adapted based on thedominant energy scales in the system. With previous approaches thatavoid the use of π pulses, a sequence that is designed to dominantlyaddress disorder would not have been possible.

Example Application: Quantum Sensing

Embodiments of the disclosed protocol can also facilitate the design ofpulse sequences for quantum sensing in the presence of interactions,disorder, and control inhomogeneities. Without being bound by theory,the sensing properties of a sequence can be captured by the Fouriertransform of the first three rows of the frame matrix F. This can alsobe conceptualized as a generalization of the usual sensing modulationfunction to the vector case. In embodiments of this disclosedapplication, the generalized modulation function describes an effectivevector magnetic field induced by the sensing signal, and highlights theimpact of both the magnitude and relative phase alignment between thecomponents in different directions.

Incorporating AC Sensing Fields into the Protocol:

without being bound by theory, AC sensing can be incorporated into thedisclosed protocol by adding an additional Hamiltonian termH _(s)(t)=B(t;ω,α)S ^(z),  (40)with the target AC sensing magnetic field chosen to be B(t; ω,α)=Re[B₀exp(−iωt+iα)], where Re denotes taking the real part, and B₀, ω,α are the amplitude, frequency and phase of the magnetic field. Theaverage Hamiltonian contribution corresponding to the sensing field canbe given by

$\begin{matrix}{{{\overset{¯}{H}}_{s} = {{\frac{1}{T}{\int_{0}^{T}{dt{B\left( {{t;\omega},\ \alpha} \right)}{\sum\limits_{\mu}{{F_{\mu}(t)}S^{\mu}}}}}} = {B_{0}{{Re}\left\lbrack {\sum\limits_{\mu}{{{\overset{˜}{F}}_{\mu}\left( {\omega,\ \alpha} \right)}S^{\mu}}} \right\rbrack}}}},} & (41)\end{matrix}$where F_(μ)(t) can satisfies U_(c)^(†)(t)S^(z)U_(c)(t)=Σ_(μ)F_(μ)(t)S^(μ), corresponding to the framematrix F_(μ,i) combined with intermediate connecting pieces due tofinite pulse durations, and

$\begin{matrix}{{{\overset{˜}{F}}_{\mu}\left( {\omega,\ \alpha} \right)} = {\left| {{\overset{˜}{F}}_{\mu}\left( {\omega,\ \alpha} \right)} \middle| e^{i{({\alpha - \phi_{\mu}})}} \right. = {\frac{1}{T}{\int_{0}^{T}{e^{{- 1}{({{\omega t} - \alpha})}}{F_{\mu}(t)}dt}}}}} & (42)\end{matrix}$is the frequency domain modulation function, which characterizes thefrequency response of a pulse sequence, according to some embodiments.

Without being bound by theory, the resulting Hamiltonian Equation (40)corresponds to a precession around an effective magnetic field specifiedby (Re[{tilde over (F)}_(x)(ω, α)], Re[{tilde over (F)}_(y)(ω, α)],Re[{tilde over (F)}_(z)(ω, α)], as illustrated in FIG. C, with totalmagnitude

$\begin{matrix}{{{{\overset{\rightarrow}{B}}_{{eff}{({\omega,\alpha})}}} = {B_{0}{{\overset{\sim}{F}}_{t}}^{2}}},{= {{B_{0}\sqrt{\sum\limits_{\mu}{{{{\overset{\sim}{F}}_{\mu}\;\left( {\omega,\ a} \right)}}^{2}{\cos^{2}\left( {\alpha - \phi_{\mu}} \right)}}}} = {{B_{0}\sqrt{\sum\limits_{\mu}{{{{\overset{\sim}{F}}_{\mu}\;\left( {\omega,\ a} \right)}}^{2}\frac{1 + {\cos\left\lbrack {2\left( {\alpha - \phi_{\mu}} \right)} \right\rbrack}}{2}}}} = {B_{0}\sqrt{{\frac{1}{2}\left\lbrack {{\sum\limits_{\mu}{{{\overset{\sim}{F}}_{\mu}\;\left( {\omega,\ a} \right)}}^{2}} + {\sum\limits_{\mu}{{Re}\left\lbrack {{\overset{\sim}{F}}_{\mu}\;\left( {\omega,\ a} \right)} \right.}^{2}}} \right\rbrack}.}}}}}} & (43)\end{matrix}$

Since {tilde over (F)}_(μ)(ω, α)² has the complex phase dependencee^(2iα), for a given target field frequency, the preceding fieldstrength has maximum value

$\begin{matrix}{\frac{\left. \max_{\alpha} \middle| {{\overset{\rightarrow}{B}}_{eff}\left( {\omega,\alpha} \right)} \right|}{B_{0}} = {\sqrt{\frac{1}{2}\left\lbrack {{\sum\limits_{\mu}{{{\overset{˜}{F}}_{\mu}\left( {\omega,\ a} \right)}}^{2}} + {{\sum\limits_{\mu}{{\overset{˜}{F}}_{\mu}(\omega)}^{2}}}} \right\rbrack}.}} & (44)\end{matrix}$Without being bound by theory, the magnetic field sensitivity will thenbe inversely proportional to the effective field strength.

Analysis of Sensing Pulse Sequences:

FIGS. 11A-11E shows example applications of the disclosed protocol foroptimal sensing sequence design. FIGS. 11A and 11B are pulse and framerepresentations of the XY-8 sequence and the fault-tolerant sensingsequence, Sequation C, respectively, according to some embodiments. AnAC sensing signal with frequency f₀ is shown below each sequence. FIG.11C shows the effective magnetic field Bar generated by the sensingsignal for a single-axis sequence (XY-8, top) and a three-axis sensingsequence (Sequation C, bottom), according to some embodiments. Withoutbeing bound by theory, the direction of B_(eff) determines the optimalinitial state for best sensitivity, in some embodiments. FIG. 11D showsa frequency domain filter function for XY-8 (left view) and Sequation C(right view), with total intensity response (upper view) and intensityfor each axis x (blue), y (red) and z (yellow), according to someembodiments. In some embodiments, the optimal sensitivity for SequationC can be achieved when all three axes act in phase (resonance {circlearound (1)} at f=f₀). If the phase is mismatched, the overallsensitivity can be reduced despite a high individual axis response(resonance {circle around (2)}). FIG. 11E is an illustration of in-phaseaddition of the response at resonance {circle around (1)}, andout-of-phase response at resonance {circle around (2)}, according tosome embodiments.

First, considering the example XY-8 sequence, employed for AC magneticfield sensing with non-interacting spins, as illustrated in FIG. 11A.Without being bound by theory, in the ideal pulse limit,F_(x)(t)=F_(y)(t)=0; meanwhile, the frame modulations in the 2-directioncan be well synchronized with an AC signal of frequency

${f_{0} = \frac{1}{2\tau}},$resulting in an effective magnetic field that is pointing in the [0, 0,1] direction, as shown in the upper view of FIG. 11C, with the maximumpossible amplitude B_(M). This is also directly illustrated in theresonance spectra of the XY-8 sequence (left view of FIG. 11D), wherethe frequency domain modulation function only has a nonzero {tilde over(F)}_(z)(ω) component (line with a peak around 1), and the othercomponents are zero.

Without being bound by theory, this type of structure where all of thesensing phase accumulation is concentrated along one axis (Z in FIG. 11Dleft view) can be incompatible with interaction-decoupling, which canrequire an equal evolution duration along each of the three axes.Redistribution of phase accumulation can be performed to address thisproblem, which can limit the effective magnetic field strength along anyaxis to be only approximately B_(M)/3.

In some embodiments of pulse sequences designed using the disclosedprotocol, utilizing all three directions of the effective field can makeit is possible to make the reduction factor only √{square root over (3)}instead, as illustrated in FIG. 11A-11E for Sequation C. In order toimprove sensitivity, this sequence can be designed with a synchronousperiodic structure along all three axes, such that an AC signal offrequency

$f_{0} = \frac{1}{2\tau}$can be aligned to efficiently accumulate phase for each of the threeaxes, as shown in FIG. 11B, in some embodiments. This can results in thefrequency domain modulation functions satisfying |{tilde over(F)}_(x)|²=|{tilde over (F)}_(y)|²=|{tilde over (F)}_(z)|² (shown in theright panel of FIG. 11D), and that the resonance peak {circle around(1)} has a total intensity |{tilde over (F)}_(t)|² that is the coherentaddition of the individual intensities along each axis (shown as peak{circle around (1)} in FIG. 11E).

Thus, although the field strength in each direction can be limited toaround B_(M)/3, use of an effective magnetic field pointing along the(1,1,1)-direction can result in a total field strength of B_(M)/√{squareroot over (3)}. This effective field direction can also use anunconventional spin initialization and readout direction, as illustratedin the lower view of FIG. 11C: instead of preparing the spin in theusual {right arrow over (x)} or ŷ axis directions, achieving high (e.g.,maximal) precession requires preparing the spin in a plane orthogonal tothe effective field direction, in some embodiments. Moreover, in someembodiments, to maximize contrast during readout, the precession planecan be rotated to contain the 2-axis, which can be equivalent torotating the precession axis into the {circumflex over (x)}−ŷ plane.Therefore, a rotation around the (1, −1,0) direction of angle arccos

$\left( \sqrt{\frac{2}{3}} \right)$can be employed for the readout pulse, while for the initializationpulse, the spin can be rotated around (−1, 1, 0) by arccos

$\left( \sqrt{\frac{2}{3}} \right)$into the (−1, −1, 2)/√{square root over (6)} direction for cosinemagnetometry, or the spin can be rotated around (1, 1, 0) by π/2 intothe (1, −1, 0) direction for sine magnetometry.

Without being bound by theory, in some embodiments, in addition to theamplitude of the frequency domain modulation function, the relativephases can also play a role in determining the resulting sensitivity.This can be the case, for example, for the two different resonance peaksin the right view of FIG. 11D. Unlike resonance {circle around (1)} withfrequency

$f = \frac{1}{2\tau}$that is illustrated in FIG. 11B, resonance {circle around (2)} hasfrequency

$f = {\frac{1}{3\tau}.}$At this frequency, the phase of the sensing signal at which precessionis maximized can be different for each of the three axes. Consequently,even though the magnitude of the example resonance peaks in each of thethree directions is the same, the terms {tilde over (F)}_(x)(ω, α),{tilde over (F)}_(y)(ω, α), {tilde over (F)}_(z)(ω, α) can be 120°out-of-phase from each other, and in Equation (44), the contribution ofthe second term is not maximized for resonance {circle around (2)}, asalso illustrated pictorially in the bottom view of FIG. 11E. Incontrast, for resonance {circle around (1)}, the signal phase that givesrise to maximal phase accumulation along each of the three axisdirections is identical, and the contributions will coherently add up.

As a result, in some embodiments, although the resonance peak heightsfor each axis direction can be slightly lower for resonance {circlearound (1)} compared to resonance {circle around (2)}, the total fieldsensitivity can be higher for the former. Consideration of both theamplitude and the phase of the frequency domain modulation functions cantherefore have an effect on sensitivity.

Without being bound by theory, in some embodiments, the precedinganalysis demonstrates that the disclosed protocol can serve as a toolfor the design of quantum sensing pulse sequences in the presence ofinteractions, disorder, and control imperfections. In addition to thescenarios explicitly described in the present disclosure, the averageHamiltonian approach can provide a characterization of other effectssuch as spurious harmonics, which cab show up as additional undesiredspectral resonances in the total modulation function.

These results can also be applicable to gradient-based magneticresonance imaging experiments with oscillating gradients. One limitationof some implementations of this approach is that at large target sensingfield amplitudes, the rotations induced by the sensing field may notcommute with the rotations designed to achieve decoupling andcancellations, and can result in a reduction of the disorder andinteraction decoupling efficiencies. The disclosed protocol can be usedto design pulse sequences that combat such issues.

Algebraic Constraints and Optimality:

Without being bound by theory, in some embodiments, optimizing formagnetic field sensitivity can also impose new algebraic constraints.The techniques described throughout the present disclosure can be usedto explore constraints on the structure of pulse sequences, as well asoptimality of sensing sequences.

First, without being bound by theory, maintaining high sensitivity forAC magnetic field sensing can be incompatible with some embodiments ofthe algebraic rules above for any pulse sequence that uses a singlepulse for π/2 rotations, and require the use of composite pulserotations to overcome. For sensing purposes and in order to decoupleon-site disorder as rapidly as possible, it can be desirable to have aperiodic structure in which the free evolution periods have framedirections that alternate between +1, −1, as shown in the example ofsequence C in FIG. 11B. However, this can imply that any interfacebetween two frame orientations can have a fixed parity, and which canlead to interaction cross terms for the case of a simple π/2transformation. Thus, in some embodiments, a composite pulse structure,in which each π/2 rotation is realized by a combination of two π/2pulses, can be utilized as shown in FIG. 6, view B.

In addition, without being bound by theory, the maximum possibleeffective field strength in the limit of ideal pulses can be determined,in some embodiments. While the effective field direction [1,1,1] can beclose to optimal, it can in fact be further improved slightly by addingan imbalance between the phase accumulation along each axis, in someembodiments. While the sum of the phase accumulation along all axes canbe fixed, the effective field strength can depend on the sum of squaresof the phase accumulation. Thus, the effective field strength can beincreased when the phase accumulation is different along the three axes,which can be achieved by choosing the frame along one of the axes to beat the troughs of the sinusoidal signal, such that there is more phaseaccumulation, in some embodiments. In the case of finite pulsedurations, unequal free evolution times along each axis can be chosen tofurther increase the imbalance and improve sensitivity.

Example Application: Quantum Simulation

While embodiments of the disclosed protocol can enable the design ofeffective dynamical decoupling sequences and optimal sensing sequences,in which the effects of disorder and interaction terms are suppressed,it can also be readily adapted to engineer various Hamiltonians.

For example, without being bound by theory, in some embodiments, whenthe conditions equations (15, 16) for decoupling different disorder andinteraction effects are not satisfied, the magnitude of the residualterm can precisely correspond to the remaining average Hamiltonian ofthe system. For example, if all other conditions are satisfied, but,Σ_(k) F_(x,k)=a≠0 then the average Hamiltonian will be

${{\overset{\sim}{H}}_{eff} = {\frac{a}{N}\Sigma_{i}\Delta_{i}S_{i}^{x}}}.$Similarly, uniform single-body terms can be engineered via, for example,intentional detuning or systematic rotation angle deviations, disorderedsingle-body terms via incomplete cancellation of onsite fields,disordered Ising interactions along each axis via incompletesymmetrization between the three axes, and/or disordered XY-typeinteractions via incomplete cancellation of cross terms.

Without being bound by theory, a few constraints can exist on the formof the two-body interaction Hamiltonian. For example, the Heisenbergcomponent of the interaction can remain invariant, since it is nottransformed under global rotations. In another example, some of theHamiltonian terms can transform in the same way (for instance disorderand imaginary spin exchange), and consequently cannot be independentlyengineered. In another example, the resultant Hamiltonian can experiencea rescaling in magnitude, due to the finite projection of the initialHamiltonian onto the final Hamiltonian. Eventually, contributions fromhigher-order terms in the Magnus expansion can also become important,but in the presence of a relatively strong target engineeredHamiltonian, higher-order terms that do not commute with the targetHamiltonian can be largely suppressed.

As a nonlimiting example of the range of Hamiltonians that may beengineered, the different possible interactions that a nativeinteraction of the form H_(int)=XX+YY−ZZ (for example, as realized inthe spin-½ subspace of interacting spin-1 NV centers) can betransformed. Without being bound by theory, in some embodiments, byevolving for a duration (1−c)T under this original Hamiltonian, andusing π/2 pulses to transform the Hamiltonian and evolve under ZZ+XX−YYand YY+ZZ−XX for duration cT/2 each, the average Hamiltonian can bewritten as{tilde over (H)} _(eff)=(1−c)(XX+YY)+(2c−1)ZZ,  (45)which depending on the value of c, can be continuously varied between anintegrable Ising interaction (c=1), a Heisenberg interaction (c=⅔), andan XY interaction (c=½). Combined with the techniques for robustengineering of other terms in the Hamiltonian, such that imperfectionsare suppressed, this facilitates access a broad range of interacting,disordered Hamiltonians, with different thermalization properties. Thus,the disclosed protocol can also serve as a powerful tool for the robustFloquet engineering of many-body Hamiltonians, for example, for quantumsimulation.

The invention claimed is:
 1. A method of reducing disorder andinteraction effects in a spin system, the method comprising: applying asequence of electromagnetic pulses to the spin system, the spin systemhaving a frame orientation in an evolution period τ₀ before a firstpulse k=1 of the sequence of electromagnetic pulses; and altering theframe orientation of the spin system with each electromagnetic pulse inthe sequence of pulses, each electromagnetic pulse being one or more ofa π/2 rotation or a π rotation, the frame orientations during thesequence conforming to the following relations:${{\sum\limits_{k}{F_{\mu k}\tau_{k}}} = 0},{and}$${{\sum\limits_{k}{{F_{xk}}\tau_{k}}} = {{\sum\limits_{k}{{F_{yk}}\tau_{k}}} = {\sum\limits_{k}{{F_{zk}}\tau_{k}}}}},$where F_(μk) represents the frame orientation of the spin system in arespective evolution period of duration τ_(k) after pulse k for eachspin direction μ=x, y, z, and where k=0 corresponds to the frameorientation F_(μ0) in the evolution period before the first pulse k=1.2. The method of claim 1, wherein the sequence of electromagnetic pulsesis periodic, and the pulses are equally spaced.
 3. The method of claim1, wherein at least one electromagnetic pulse of the sequence ofelectromagnetic pulses includes two or more π/2 rotations, and the spinsystem further comprises intermediary frame orientations representingthe frame orientation of the spin system after each but a final π/2rotation, the intermediary frame orientations conforming to thefollowing relations:${{\sum\limits_{k}F_{\mu\; v}} = 0},{{\sum\limits_{k}{F_{xv}}} = {{\sum\limits_{k}{F_{yv}}} = {\sum\limits_{k}{F_{zv}}}}}$where F_(μv) represents the frame orientation of the spin system foreach intermediary frame v for each spin direction μ=x, y, z.
 4. Themethod of claim 1, wherein for each π rotation, the frame orientation ofthe spin system further comprises an intermediary frame orientationrepresenting the frame orientation of the spin system after the firstπ/2 rotation of the π rotation, intermediary frame orientations togetherconforming to the following relations:${{\sum\limits_{k}F_{\mu\; v}} = 0},{{\sum\limits_{k}{F_{xv}}} = {{\sum\limits_{k}{F_{yv}}} = {\sum\limits_{k}{F_{zv}}}}}$where F_(μv) represents the frame orientation of the spin system foreach intermediary frame v for each spin direction μ=x, y, z.
 5. Themethod of claim 1, wherein for each pair of axes μ,μ=x, y, z, the parityof frame changes sums to zero such that${{{\sum\limits_{k}{F_{\mu,k}F_{v,{k + 1}}}} + {F_{\mu,\;{k + 1}}F_{v,k}}} = 0},$for (μ, v)=(x, y), (x, z), (y, z).
 6. The method of claim 1, wherein thechirality of frame changes sums to zero such that the cyclic sum${{\sum\limits_{k}{{\overset{->}{F}}_{k} \times {\overset{\rightarrow}{F}}_{k + 1}}} = \overset{\rightarrow}{0}},$where {circumflex over (F)}_(k)=Σ_(u)F_(μ,k) {right arrow over (e)}_(μ)and {right arrow over (e)}_(μ) are the unit vectors along axisdirections.
 7. The method of claim 1, further comprising: generating aneffective magnetic field {right arrow over (B)}_(eff) as seen by thedriven spins; and initializing the frame orientation of the spin systemto be perpendicular to the effective magnetic field.
 8. The method ofclaim 1, wherein the sequence of electromagnetic pulses is used toincrease the coherence time of an ensemble of nitrogen-vacancy (NV)centers in diamond beyond a spin-spin interaction sensitivity limit. 9.The method of claim 1, wherein the sequence of electromagnetic pulses isused to increase the coherence time of a magnetic field sensing ensembleof nitrogen-vacancy (NV) centers in diamond such that a sensitivity ofthe magnetic field sensing ensemble of NV centers overcomes a spin-spininteraction sensitivity limit.
 10. A system, comprising: a quantumsensor comprising an ensemble of spins in solid state, the ensemble ofspins having a density in which the interactions between the spins limita coherence time of the ensemble of spins in solid state; and a pulsegenerator configured to apply electromagnetic radiation to the quantumsensor, the electromagnetic radiation decoupling the interactionsbetween the spins to increase the coherence time beyond a spin-spininteraction sensitivity limit of the ensemble of spins when measuring atarget signal.
 11. The system of claim 10, wherein the quantum sensorcomprising an ensemble of NV centers in diamond of density r ppm, andthe coherence time is increased to be longer than a value of 72/r us (asdetermined from the scaling of the interaction limit), up to 1 ms. 12.The system of claim 10, wherein the pulse generator applieselectromagnetic radiation to the quantum sensor according to the methodof claim
 1. 13. A system comprising: a spin system; and a pulsegenerator configured to a sequence of electromagnetic pulses to the spinsystem, the spin system having a frame orientation in an evolutionperiod τ₀ before a first pulse k=1 of the sequence of electromagneticpulses, each electromagnetic pulse corresponding to a frame of thesequence of pulses, and each electromagnetic pulse being one or more ofa π/2 rotation or a π rotation, the frame orientations during thesequence conforming to the following relations:${{\sum\limits_{k}{F_{\mu k}\tau_{k}}} = 0},{and}$${{\sum\limits_{k}{{F_{xk}}\tau_{k}}} = {{\sum\limits_{k}{{F_{yk}}\tau_{k}}} = {\sum\limits_{k}{{F_{zk}}\tau_{k,}}}}},$where F_(μk) represents the frame orientation of the spin system in arespective evolution period of duration τ_(k) after pulse k for eachspin direction μ=x, y, z, and where k=0 corresponds to the frameorientation F_(μ0) in the evolution period before the first pulse k=1.14. The system of claim 13, wherein the sequence of electromagneticpulses is periodic, and the pulses are equally spaced.
 15. The system ofclaim 13, wherein at least one electromagnetic pulse of the sequence ofelectromagnetic pulses includes two or more π/2 rotations, and the spinsystem further comprises intermediary frame orientations representingthe frame orientation of the spin system after each but a final π/2rotation, the intermediary frame orientations conforming to thefollowing relations:${{\sum\limits_{k}F_{\mu\; v}} = 0},{{\sum\limits_{k}{F_{xv}}} = {{\sum\limits_{k}{F_{yv}}} = {\sum\limits_{k}{F_{zv}}}}}$where F_(μv) represents the frame orientation of the spin system foreach intermediary frame v for each spin direction μ=x, y, z.
 16. Thesystem of claim 13, wherein for each π rotation, the frame orientationof the spin system further comprises an intermediary frame orientationrepresenting the frame orientation of the spin system after the firstπ/2 rotation of the π rotation, intermediary frame orientations togetherconforming to the following relations:${{\sum\limits_{k}F_{\mu\; v}} = 0},{{\sum\limits_{k}{F_{xv}}} = {{\sum\limits_{k}{F_{yv}}} = {\sum\limits_{k}{F_{zv}}}}}$where F_(μv) represents the frame orientation of the spin system foreach intermediary frame v for each spin direction μ=x, y, z.
 17. Thesystem of claim 13, wherein for each pair of axes μ,μ=x, y, z, parity offrame changes experienced by the spin system sums to zero such that${{{\sum\limits_{k}{F_{\mu,k}F_{v,{k + 1}}}} + {F_{\mu\;,{k + 1}}F_{v,k}}} = 0},$for (μ, v)=(x, y), (x, z), (y, z).
 18. The system of claim 13, whereinchirality of frame changes experienced by the spin system sums to zerosuch that the cyclic sum${{\sum\limits_{k}{{\overset{->}{F}}_{k} \times {\overset{\rightarrow}{F}}_{k + 1}}} = \overset{\rightarrow}{0}},$where {circumflex over (F)}_(k)=Σ_(μ)F_(μ,k){right arrow over (e)}_(μ)and {right arrow over (e)}_(μ) are the unit vectors along axisdirections.